🎯⭐ INTERACTIVE LESSON

Trigonometric Identities

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Trigonometric Identities - Complete Interactive Lesson

Part 1: Pythagorean Identities

🔗 Trigonometric Identities — Pythagorean Identities

Part 1 of 7

The Pythagorean identities are the foundation of all trig simplification. They come from dividing $\sin^2\theta + \cos^2\theta = 1$ by different functions.

The Three Pythagorean Identities

Identity	Derived by	Most useful when...
$\boxed{\sin^2\theta + \cos^2\theta = 1}$

Useful Rearrangements

Form	Rearrangement
$\sin^2\theta =$	$1 - \cos^2\theta$

📝 Worked Examples

Example 1: Simplify $\frac{1 - \cos^2\theta}{\sin\theta}$

Replace with :

🎯 Simplification Strategy

Decision Tree for Pythagorean Identities

See this in the expression...	Replace with...
$\sin^2\theta + \cos^2\theta$	$1$

Concept Check 🎯

Pythagorean Identity Practice 🧮

1) Simplify $\frac{\cos^2\theta}{1 - \sin^2\theta}$ . Write the simplified result. (e.g., )

Identity Matching 🔽

Exit Quiz ✅

Part 2: Sum & Difference Formulas

🔄 Trigonometric Identities — Reciprocal & Quotient Identities

Part 2 of 7

The reciprocal and quotient identities rewrite all six trig functions in terms of sine and cosine — the most powerful simplification strategy.

Reciprocal Identities

$\boxed{\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}}$

Part 3: Double-Angle Formulas

🪞 Trigonometric Identities — Even-Odd & Cofunction Identities

Part 3 of 7

Even-odd identities describe what happens when you negate an angle. Cofunction identities link a function to its complement. Both are shortcuts for rewriting expressions without a calculator.

Even-Odd Identities

Function	$f(-\theta)$	Type
$\cos(-\theta)$	$\cos\theta$

Part 4: Half-Angle Formulas

➕ Trigonometric Identities — Sum & Difference Formulas

Part 4 of 7

The sum and difference identities let you expand $\sin(A \pm B)$ , $\cos(A \pm B)$ , and $\tan(A \pm B)$ into expressions involving only , , , .

Part 5: Verifying Identities

✖️ Trigonometric Identities — Double-Angle & Half-Angle Formulas

Part 5 of 7

Double-angle formulas express $\sin 2\theta$ , $\cos 2\theta$ , and $\tan 2\theta$ in terms of functions of $\theta$ . Half-angle formulas go the other direction: expressing , etc., in terms of .

Part 6: Problem-Solving Workshop

✅ Trigonometric Identities — Verifying Identities

Part 6 of 7

Verifying (or proving) a trigonometric identity means showing that the left side equals the right side for all values in the domain. You never cross-multiply or move terms across the equals sign — you work one side only until it matches the other.

The Golden Rules

Rule	Why
Work one side only	An identity is not an equation to "solve" — you must transform, not rearrange
Start with the more complex side	More terms = more opportunities to simplify
Convert everything to sin and cos	Common denominators and cancellations become visible
Factor when possible	$\sin^2\theta - \cos^2\theta$ factors as

Part 7: Review & Applications

🧩 Trigonometric Identities — Full Synthesis

Part 7 of 7

This final part combines every identity type from Parts 1–6 into mixed problems. The challenge: recognizing which identity to apply and when.

Complete Identity Reference

Category	Key Formulas
Pythagorean	$\sin^2\theta + \cos^2\theta = 1$ , ,

$\frac{1 - \cos^2\theta}{\sin\theta} = \frac{\sin^2\theta}{\sin\theta} = \sin\theta$

Example 2: Simplify $\sec^2\theta - \tan^2\theta$

From $1 + \tan^2\theta = \sec^2\theta$ , rearrange:

$\sec^2\theta - \tan^2\theta = 1$

This is always $1$ , regardless of $\theta$ . Great exam shortcut!

Example 3: Express $\tan^2\theta$ in terms of $\cos\theta$ only

$\tan^2\theta = \sec^2\theta - 1 = \frac{1}{\cos^2\theta} - 1 = \frac{1 - \cos^2\theta}{\cos^2\theta} = \frac{\sin^2\theta}{\cos^2\theta}$

Example 4: Given $\sin\theta = \frac{2}{3}$ (Q I), find all six trig values

Step	Computation
$\cos\theta$	$\sqrt{1 - 4/9} = \frac{\sqrt{5}}{3}$
$\tan\theta$	$\frac{2/3}{\sqrt{5}/3} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$
$\csc\theta$	$\frac{3}{2}$
$\sec\theta$	$\frac{3}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$
$\cot\theta$	$\frac{\sqrt{5}}{2}$

Pro tip: When you see a sum or difference involving squared trig functions and the number $1$ , a Pythagorean identity is almost certainly the key.

\frac{\sin^2\theta}{1 - \cos^2\theta} = \frac{\sin^2\theta}{\sin^2\theta} = 1

2) If $\sec\theta = \frac{5}{4}$ and $\theta$ is in Q I, find $\tan\theta$ . Write as a fraction. (e.g., $\sec\alpha = \frac{13}{5}$ : $\tan^2\alpha = 169/25 - 1 = 144/25$ , so $\tan\alpha = 12/5$ )

3) Simplify $\sin^2\theta \cdot \csc^2\theta + \cos^2\theta \cdot \sec^2\theta$ . Write as an integer. (e.g., $\sin\theta \cdot \csc\theta = 1$ since they're reciprocals)

$\boxed{\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}}$

Why Convert to Sine & Cosine?

Advantage	Example
Common denominator	$\tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}$ → combine
Cancel factors	$\sin\theta \cdot \csc\theta = \sin\theta \cdot \frac{1}{\sin\theta} = 1$
Reveal Pythagorean forms	$\sec^2\theta = \frac{1}{\cos^2\theta}$ connects to

📝 Worked Examples

Example 1: Simplify $\tan\theta \cdot \cos\theta$

$\tan\theta \cdot \cos\theta = \frac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta$

Example 2: Simplify $\frac{\cot\theta}{\csc\theta}$

$\frac{\cot\theta}{\csc\theta} = \frac{\cos\theta/\sin\theta}{1/\sin\theta} = \frac{\cos\theta}{\sin\theta} \cdot \frac{\sin\theta}{1} = \cos\theta$

Example 3: Simplify $\tan\theta + \cot\theta$

$\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta} = \csc\theta\sec\theta$

Example 4: Simplify $\sec\theta - \cos\theta$

$\frac{1}{\cos\theta} - \cos\theta = \frac{1 - \cos^2\theta}{\cos\theta} = \frac{\sin^2\theta}{\cos\theta} = \sin\theta \cdot \tan\theta$

🎯 The "Convert Everything" Strategy

Step-by-Step Process

Step	Action	Example
1	Replace $\tan, \cot, \sec, \csc$ with $\sin/\cos$	$\sec\theta \to \frac{1}{\cos\theta}$
2	Find a common denominator	Combine fractions
3	Simplify numerator using Pythagorean identities	$\sin^2 + \cos^2 = 1$
4	Cancel common factors	Reduce the fraction
5	Convert back if a cleaner form exists	$\frac{\sin\theta}{\cos\theta} \to \tan\theta$

Common Products That Equal 1

Product	Why
$\sin\theta \cdot \csc\theta$	$= \sin\theta \cdot \frac{1}{\sin\theta} = 1$

Concept Check 🎯

Simplification Practice 🧮

1) Simplify $\frac{\sec\theta}{\csc\theta}$ . Write as a single trig function (e.g., sin, cos, tan, cot, sec, csc). (e.g., $\frac{\csc\theta}{\sec\theta} = \frac{1/\sin\theta}{1/\cos\theta} = \frac{\cos\theta}{\sin\theta} = \cot\theta$ )

2) Simplify $\cot\theta \cdot \sin\theta$ . Write as a single trig function. (e.g., $\tan\theta \cdot \cos\theta = \frac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta$ )

3) Evaluate $\tan 60° \cdot \cot 60°$ . Write as an integer. (e.g., $\sin 45° \cdot \csc 45° = 1$ since they are reciprocals)

Identity Matching 🔽

Memory aid: Only cosine and secant are even — the "co-s" pair. Everything else is odd.

Cofunction Identities (Complementary Angles)

$\boxed{\sin\theta = \cos\!\left(\frac{\pi}{2} - \theta\right) \qquad \cos\theta = \sin\!\left(\frac{\pi}{2} - \theta\right)}$

$\tan\theta = \cot\!\left(\frac{\pi}{2} - \theta\right) \qquad \sec\theta = \csc\!\left(\frac{\pi}{2} - \theta\right)$

The co in cosine, cosecant, cotangent stands for complement!

📝 Worked Examples

Example 1: Simplify $\sin(-\theta)\cos(-\theta)$

$\sin(-\theta)\cos(-\theta) = (-\sin\theta)(\cos\theta) = -\sin\theta\cos\theta$

Sine is odd (picks up a negative), cosine is even (stays the same).

Example 2: Evaluate $\cos(-60°)$ without a calculator

$\cos$ is even, so $\cos(-60°) = \cos 60° = \frac{1}{2}$ .

Example 3: Rewrite $\sin 70°$ as a cosine

$\sin 70° = \cos(90° - 70°) = \cos 20°$

Example 4: Show that $\tan(-\theta) + \cot(90° - \theta)$ simplifies to $0$

$\tan(-\theta) + \cot(90° - \theta) = -\tan\theta + \tan\theta = 0$

The cofunction identity gives $\cot(90° - \theta) = \tan\theta$ , and the even-odd identity gives $\tan(-\theta) = -\tan\theta$ .

🔍 Why These Work — Unit Circle Reasoning

Even-Odd: Reflection Across the $x$ -axis

Negating $\theta$ reflects the point $(\cos\theta,\,\sin\theta)$ to $(\cos\theta,\,-\sin\theta)$ .

Coordinate	After Reflection	Conclusion
$x$ -coordinate ( $\cos$ )	Unchanged	$\cos(-\theta) = \cos\theta$ — even
-coordinate ()

Cofunction: $90°$ Rotation

The point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$ .

The point at angle $\frac{\pi}{2} - \theta$ has coordinates $(\sin\theta, \cos\theta)$ — the and swap!

This swap is exactly why $\sin\theta = \cos(90° - \theta)$ .

Quick Decision Table

I want to …	Use …
Remove a negative angle	Even-odd identities
Replace $\sin$ with $\cos$ (or vice versa)	Cofunction identities
Both at once	Chain them: even-odd first, cofunction second

Concept Check 🎯

Even-Odd & Cofunction Practice 🧮

1) $\cos(-120°) = \cos\,\_\_°$ . Write the positive angle in degrees. (e.g., $\cos(-45°) = \cos 45°$ since cosine is even)

2) $\sin 25° = \cos\,\_\_°$ . Write the complementary angle in degrees. (e.g., $\sin 40° = \cos 50°$ since $40 + 50 = 90$ )

3) Evaluate $\tan(-45°)$ . Write as an integer. (e.g., $\sin(-30°) = -\sin 30° = -1/2$ since sine is odd)

Classification & Matching 🔽

The Big Six Formulas

$\boxed{\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B}$

$\boxed{\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B}$

$\boxed{\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}}$

Sign Pattern Summary

Formula	Plus version	Minus version
$\sin(A \pm B)$	same sign ( $+$ )	same sign ( $-$ )
$\cos(A \pm B)$	opposite sign ( $-$ )	opposite sign ( $+$ )
$\tan(A \pm B)$	numerator $+$ , denominator $-$	numerator $-$ , denominator $+$

Memory aid for cosine: "Cosine is contrary" — the sign in the formula is opposite the sign in the argument.

📝 Worked Examples

Example 1: Find the exact value of $\cos 75°$

Split: $75° = 45° + 30°$

$\cos 75° = \cos 45°\cos 30° - \sin 45°\sin 30°$

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Example 2: Find the exact value of $\sin 15°$

Split: $15° = 45° - 30°$

$\sin 15° = \sin 45°\cos 30° - \cos 45°\sin 30°$

$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$

Example 3: Simplify $\sin(x + \pi)$

$\sin(x + \pi) = \sin x\cos\pi + \cos x\sin\pi = \sin x(-1) + \cos x(0) = -\sin x$

This confirms the identity: shifting by $\pi$ negates sine.

Example 4: Find $\tan 75°$

$\tan(45° + 30°) = \frac{\tan 45° + \tan 30°}{1 - \tan 45°\tan 30°} = \frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}} = 2 + \sqrt{3}$

🎯 Choosing the Right Angle Decomposition

Common Angle Splits

Target Angle	Split As	Using
$15°$	$45° - 30°$	Difference
$75°$	$45° + 30°$	Sum
$105°$	$60° + 45°$	Sum
$165°$	$180° - 15°$	or $120° + 45°$
$\frac{\pi}{12}$	$\frac{\pi}{4} - \frac{\pi}{6}$
$\frac{5\pi}{12}$	$\frac{\pi}{4} + \frac{\pi}{6}$
$\frac{7\pi}{12}$	$\frac{\pi}{3} + \frac{\pi}{4}$

When to Use Sum/Difference Formulas

Situation	Example
Exact value of a non-standard angle	$\sin 75°$ , $\cos 15°$
Expression has $\sin A\cos B \pm \cos A\sin B$

Concept Check 🎯

Exact Value Computation 🧮

1) Find the exact value of $\cos 15°$ . The answer has the form $\frac{\sqrt{a}+\sqrt{b}}{4}$ . What is $a + b$ ? (e.g., if the answer were $\frac{\sqrt{5}+\sqrt{3}}{4}$ , you'd enter $8$ )

2) Simplify $\cos(x + 2\pi)$ . Write as a single trig function of $x$ (e.g., sin, cos, tan). (e.g., $\sin(x + 2\pi) = \sin x$ by periodicity)

3) Evaluate $\sin 45°\cos 15° + \cos 45°\sin 15°$ . This matches $\sin(A+B)$ ; enter the result as a fraction. (e.g., $\sin$ )

Formula Matching 🔽

\sin\frac{\theta}{2}

Double-Angle Identities

$\boxed{\sin 2\theta = 2\sin\theta\cos\theta}$

$\boxed{\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta}$

$\boxed{\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}}$

Why three forms for $\cos 2\theta$ ? Each is best in different situations:

Form	Best when you know …
$\cos^2\theta - \sin^2\theta$	Both $\sin\theta$ and $\cos\theta$
$2\cos^2\theta - 1$	Only $\cos\theta$
$1 - 2\sin^2\theta$	Only $\sin\theta$

Half-Angle Identities

$\boxed{\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}} \qquad \boxed{\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}}$

$\boxed{\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}}$

The $\pm$ depends on the quadrant of $\frac{\theta}{2}$ , not of $\theta$ !

📝 Worked Examples

Example 1: Given $\sin\theta = \frac{3}{5}$ with $\theta$ in QI, find $\sin 2\theta$

Since $\sin\theta = 3/5$ and QI: $\cos\theta = 4/5$ .

$\sin 2\theta = 2\sin\theta\cos\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

Example 2: Find $\cos 2\theta$ given $\cos\theta = -\frac{1}{3}$

Use the form that only needs $\cos\theta$ :

$\cos 2\theta = 2\cos^2\theta - 1 = 2\left(\frac{1}{9}\right) - 1 = \frac{2}{9} - 1 = -\frac{7}{9}$

Example 3: Find the exact value of $\sin 15°$ using the half-angle formula

$15° = \frac{30°}{2}$ , so $\theta = 30°$ and $\cos 30 ° = \frac{}{}$ .

$\sin 15° = +\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2}$

( $+$ because $15°$ is in QI)

Example 4: Power-Reduction Formula

The $\cos 2\theta$ identity rearranges to eliminate squares:

$\sin^2\theta = \frac{1 - \cos 2\theta}{2} \qquad \cos^2\theta = \frac{1 + \cos 2\theta}{2}$

These are essential for calculus integration of $\sin^2 x$ and $\cos^2 x$ .

🔗 Where Double-Angle Comes From

Double-angle formulas are just the sum formulas with $B = A$ :

$\sin(A + A) = \sin A\cos A + \cos A\sin A = 2\sin A\cos A$

$\cos(A + A) = \cos A\cos A - \sin A\sin A = \cos^2 A - \sin^2 A$

Decision Flowchart

I see …	I should …
$\sin\theta\cos\theta$	Use $\sin 2\theta = 2\sin\theta\cos\theta$
or alone

Concept Check 🎯

Double-Angle Computation 🧮

1) If $\sin\theta = \frac{5}{13}$ and $\cos\theta = \frac{12}{13}$ , find $\sin 2\theta$ . Write as a fraction. (e.g., if $\sin\theta = 3/5, \cos\theta = 4/5$ , then $\sin 2\theta = 2(3/5)(4/5) = 24/25$ )

2) Find $\cos 2\theta$ if $\sin\theta = \frac{1}{4}$ . Write as a fraction. (e.g., $\cos 2 θ = 1 -$ )

3) If $\cos\theta = \frac{3}{5}$ and $\sin\theta = \frac{4}{5}$ , find . Write as a fraction. (e.g., with , is undefined)

Formula Recognition 🔽

(\sin\theta - \cos\theta)(\sin\theta + \cos\theta)

📝 Worked Verifications

Verify: $\frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$

Strategy: Work the left side. Multiply by the conjugate $\frac{1 - \cos\theta}{1 - \cos\theta}$ :

$\frac{\sin\theta}{1 + \cos\theta} \cdot \frac{1 - \cos\theta}{1 - \cos\theta} = \frac{\sin\theta(1 - \cos\theta)}{1 - \cos^2\theta} = \frac{\sin\theta(1 - \cos\theta)}{\sin^2\theta} = \frac{1 - \cos\theta}{\sin\theta} \;\checkmark$

Verify: $\tan\theta + \cot\theta = \sec\theta\csc\theta$

Strategy: Convert the left side to sin/cos:

$\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta} = \sec\theta\csc\theta \;\checkmark$

Verify: $\frac{1 + \sin\theta}{\cos\theta} = \frac{\cos\theta}{1 - \sin\theta}$

Strategy: Cross-reference by working the right side — multiply by conjugate $\frac{1 + \sin\theta}{1 + \sin\theta}$ :

$\frac{\cos\theta}{1 - \sin\theta} \cdot \frac{1 + \sin\theta}{1 + \sin\theta} = \frac{\cos\theta(1 + \sin\theta)}{1 - \sin^2\theta} = \frac{\cos\theta(1 + \sin\theta)}{\cos^2\theta} = \frac{1 + \sin\theta}{\cos\theta} \;\checkmark$

🛠️ Verification Toolkit — Decision Flowchart

Which Strategy Do I Use?

I see …	Try …
Fractions on one side	Combine into a single fraction
$1 \pm \sin\theta$ or $1 \pm \cos\theta$ in a denominator	Multiply by the conjugate
$\sec, \csc, \tan, \cot$	Convert to $\sin$ and $\cos$
Squares like $\sin^2\theta$ or $\cos^2\theta$	Apply Pythagorean identity
$\sin^2\theta - \cos^2\theta$ or similar	Factor as a difference of squares
$\sin 2\theta$ or $\cos 2\theta$	Expand using double-angle formulas
Nothing obvious	Try both sides and see which simplifies to a recognizable form

Common Mistakes to Avoid

Mistake	Why It's Wrong
Moving terms across the $=$ sign	You're proving equality, not solving
Working both sides toward a "common middle"	Only acceptable if you work each side independently
Dividing both sides by a trig expression	Not allowed — it's not an equation
Stopping before the sides match exactly	The transformed side must be identical to the target

Concept Check 🎯

Verification Computation 🧮

1) In verifying $\tan\theta + \cot\theta = \sec\theta\csc\theta$ , the combined left side has numerator $\sin^2\theta + \cos^2\theta$ . This simplifies to what integer? (e.g., the numerator $a^2 - a^2$ simplifies to $0$ )

2) To verify $\frac{\sin\theta}{1+\cos\theta} = \frac{1-\cos\theta}{\sin\theta}$ , you multiply the left fraction by . The new denominator equals . What is ? (e.g., might become , so )

3) In the identity $\sec\theta - \cos\theta = \sin\theta\tan\theta$ , converting the left side gives $\frac{1 - \cos^2\theta}{\cos\theta}$ . The numerator becomes . What is ? (e.g., might yield , so )

Strategy Matching 🔽

1 + \tan^2\theta = \sec^2\theta

1 + \cot^2\theta = \csc^2\theta

🗺️ Identity Selection Flowchart

What Do I See? → What Do I Use?

Pattern in Expression	Identity to Apply
$\sin^2$ or $\cos^2$ alone	Pythagorean → replace with $1 - \text{other}^2$
$\sec, \csc, \tan, \cot$ mixed	Reciprocal/Quotient → convert to sin/cos
Negative angle $(-\theta)$	Even-odd
$90° - \theta$ or $\frac{\pi}{2} - \theta$	Cofunction
Non-standard angle ( $15°, 75°, 105°$ …)	Sum/Difference formulas
$\sin\theta\cos\theta$ product	Double-angle: $= \frac{1}{2}\sin 2\theta$
$\cos^2\theta - \sin^2\theta$	Recognize $= \cos 2\theta$
$1 \pm \cos\theta$ in denominator	Conjugate multiply, or half-angle
Verifying LHS = RHS	Work the complex side only; never cross the $=$

Multi-Step Strategy

Scan — Identify the identity types present
Convert — Rewrite everything in sin/cos if mixed functions appear
Combine — Get a single fraction if multiple terms
Substitute — Apply Pythagorean, double-angle, etc.
Simplify — Cancel and reduce

📝 Mixed Worked Examples

Example 1: Simplify $\frac{\sin 2\theta}{1 + \cos 2\theta}$

Use double-angle expansions:

Numerator: $\sin 2\theta = 2\sin\theta\cos\theta$
Denominator: $1 + \cos 2\theta = 1 + (2\cos^2\theta - 1) = 2\cos^2\theta$

$\frac{2\sin\theta\cos\theta}{2\cos^2\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta$

Example 2: Find $\sin 75°\cos 15° + \cos 75°\sin 15°$

Recognize the sum pattern: $\sin A\cos B + \cos A\sin B = \sin(A + B)$

$= \sin(75° + 15°) = \sin 90° = 1$

Example 3: Simplify $\frac{\sec(-\theta)}{\csc(90° - \theta)}$

Apply even-odd: $\sec(-\theta) = \sec\theta$ (even).

Apply cofunction: $\csc(90° - \theta) = \sec\theta$ .

$\frac{\sec\theta}{\sec\theta} = 1$

Example 4: Verify $\frac{\sin 2\theta}{\sin\theta} - \frac{\cos 2\theta}{\cos\theta} = \sec\theta$

Work the left side:

$\frac{2\sin\theta\cos\theta}{\sin\theta} - \frac{1 - 2\sin^2\theta}{\cos\theta} = 2\cos\theta - \frac{1 - 2\sin^2\theta}{\cos\theta}$

$= \frac{2\cos^2\theta - 1 + 2\sin^2\theta}{\cos\theta} = \frac{2(\cos^2\theta + \sin^2\theta) - 1}{\cos\theta} = \frac{2 - 1}{\cos\theta} = \frac{1}{\cos\theta} = \sec\theta \;\checkmark$

Mixed Identity Quiz 🎯

Cross-Topic Computation 🧮

1) $\cos^2 15° + \sin^2 15°$ = ? Write as an integer. (e.g., $\cos^2 73° + \sin^2 73° = 1$ by Pythagorean identity)

2) Simplify $\frac{\sin 2\theta}{2\sin\theta}$ to a single trig function. Write the function name. (e.g., $\frac{\cos 2\theta + 1}{2\cos\theta}$ simplifies by expanding )

3) $\sin(-30°)\sec(-30°)$ = ? Write as a fraction. (e.g., $\cos(-60°)\csc(-60°) = \cos 60° \cdot (-\csc 60°) = \frac{1}{2} \cdot (-\frac{2\sqrt{3}}{3})$ )

Identity Classification 🔽

Final Exit Quiz ✅

sinθcosθsin2θ+cos2θ=

cos 30° = \frac{3}{2}

cos 2 θ = 1 - 2 (3/5)^{2} = 1 - 18/25 = 7/25

\tan 2\theta = \frac{2(1)}{1-1^2}

1−cos2θsinθ(1−cosθ)=

sin2θsinθ(1−cosθ)=

sinθcosθsin2θ+cos2θ=

1−sin2θcosθ(1+sinθ)=

cos2θcosθ(1+sinθ)=

\frac{1-\cos\theta}{1-\cos\theta}

cosθ2(cos2θ+sin2θ)−1=

Trigonometric Identities

Trigonometric Identities - Complete Interactive Lesson

Part 1: Pythagorean Identities

🔗 Trigonometric Identities — Pythagorean Identities

The Three Pythagorean Identities

Useful Rearrangements

📝 Worked Examples

Example 1: Simplify 1−cos⁡2θsin⁡θ\frac{1 - \cos^2\theta}{\sin\theta}sinθ1−cos2θ​

🎯 Simplification Strategy

Decision Tree for Pythagorean Identities

Part 2: Sum & Difference Formulas

🔄 Trigonometric Identities — Reciprocal & Quotient Identities

Reciprocal Identities

Part 3: Double-Angle Formulas

🪞 Trigonometric Identities — Even-Odd & Cofunction Identities

Even-Odd Identities

Part 4: Half-Angle Formulas

➕ Trigonometric Identities — Sum & Difference Formulas

Part 5: Verifying Identities

✖️ Trigonometric Identities — Double-Angle & Half-Angle Formulas

Part 6: Problem-Solving Workshop

✅ Trigonometric Identities — Verifying Identities

The Golden Rules

Part 7: Review & Applications

🧩 Trigonometric Identities — Full Synthesis

Complete Identity Reference

Example 2: Simplify sec⁡2θ−tan⁡2θ\sec^2\theta - \tan^2\thetasec2θ−tan2θ

Example 3: Express tan⁡2θ\tan^2\thetatan2θ in terms of cos⁡θ\cos\thetacosθ only

Example 4: Given sin⁡θ=23\sin\theta = \frac{2}{3}sinθ=32​ (Q I), find all six trig values

Quotient Identities

Why Convert to Sine & Cosine?

📝 Worked Examples

Example 1: Simplify tan⁡θ⋅cos⁡θ\tan\theta \cdot \cos\thetatanθ⋅cosθ

Example 2: Simplify cot⁡θcsc⁡θ\frac{\cot\theta}{\csc\theta}cscθcotθ​

Example 3: Simplify tan⁡θ+cot⁡θ\tan\theta + \cot\thetatanθ+cotθ

Example 4: Simplify sec⁡θ−cos⁡θ\sec\theta - \cos\thetasecθ−cosθ

🎯 The "Convert Everything" Strategy

Step-by-Step Process

Common Products That Equal 1

Cofunction Identities (Complementary Angles)

📝 Worked Examples

Example 1: Simplify sin⁡(−θ)cos⁡(−θ)\sin(-\theta)\cos(-\theta)sin(−θ)cos(−θ)

Example 2: Evaluate cos⁡(−60°)\cos(-60°)cos(−60°) without a calculator

Example 3: Rewrite sin⁡70°\sin 70°sin70° as a cosine

Example 4: Show that tan⁡(−θ)+cot⁡(90°−θ)\tan(-\theta) + \cot(90° - \theta)tan(−θ)+cot(90°−θ) simplifies to 000

🔍 Why These Work — Unit Circle Reasoning

Even-Odd: Reflection Across the xxx-axis

Cofunction: 90°90°90° Rotation

Quick Decision Table

The Big Six Formulas

Sign Pattern Summary

📝 Worked Examples

Example 1: Find the exact value of cos⁡75°\cos 75°cos75°

Example 2: Find the exact value of sin⁡15°\sin 15°sin15°

Example 3: Simplify sin⁡(x+π)\sin(x + \pi)sin(x+π)

Example 4: Find tan⁡75°\tan 75°tan75°

🎯 Choosing the Right Angle Decomposition

Common Angle Splits

When to Use Sum/Difference Formulas

Double-Angle Identities

Half-Angle Identities

📝 Worked Examples

Example 1: Given sin⁡θ=35\sin\theta = \frac{3}{5}sinθ=53​ with θ\thetaθ in QI, find sin⁡2θ\sin 2\thetasin2θ

Example 2: Find cos⁡2θ\cos 2\thetacos2θ given cos⁡θ=−13\cos\theta = -\frac{1}{3}cosθ=−31​

Example 3: Find the exact value of sin⁡15°\sin 15°sin15° using the half-angle formula

Example 4: Power-Reduction Formula

🔗 Where Double-Angle Comes From

Decision Flowchart

📝 Worked Verifications

Verify: sin⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ\frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}1+cosθsinθ​=sinθ1−cosθ​

Verify: tan⁡θ+cot⁡θ=sec⁡θcsc⁡θ\tan\theta + \cot\theta = \sec\theta\csc\thetatanθ+cotθ=secθcscθ

Verify: 1+sin⁡θcos⁡θ=cos⁡θ1−sin⁡θ\frac{1 + \sin\theta}{\cos\theta} = \frac{\cos\theta}{1 - \sin\theta}cosθ1+sinθ​=

🛠️ Verification Toolkit — Decision Flowchart

Which Strategy Do I Use?

Common Mistakes to Avoid

🗺️ Identity Selection Flowchart

What Do I See? → What Do I Use?

Multi-Step Strategy

📝 Mixed Worked Examples

Example 1: Simplify sin⁡2θ1+cos⁡2θ\frac{\sin 2\theta}{1 + \cos 2\theta}1+cos2θsin2θ​

Example 1: Simplify $\frac{1 - \cos^2\theta}{\sin\theta}$

Example 2: Simplify $\sec^2\theta - \tan^2\theta$

Example 3: Express $\tan^2\theta$ in terms of $\cos\theta$ only

Example 4: Given $\sin\theta = \frac{2}{3}$ (Q I), find all six trig values

Example 1: Simplify $\tan\theta \cdot \cos\theta$

Example 2: Simplify $\frac{\cot\theta}{\csc\theta}$

Example 3: Simplify $\tan\theta + \cot\theta$

Example 4: Simplify $\sec\theta - \cos\theta$

Example 1: Simplify $\sin(-\theta)\cos(-\theta)$

Example 2: Evaluate $\cos(-60°)$ without a calculator

Example 3: Rewrite $\sin 70°$ as a cosine

Example 4: Show that $\tan(-\theta) + \cot(90° - \theta)$ simplifies to $0$

Even-Odd: Reflection Across the $x$ -axis

Cofunction: $90°$ Rotation

Example 1: Find the exact value of $\cos 75°$

Example 2: Find the exact value of $\sin 15°$

Example 3: Simplify $\sin(x + \pi)$

Example 4: Find $\tan 75°$

Example 1: Given $\sin\theta = \frac{3}{5}$ with $\theta$ in QI, find $\sin 2\theta$

Example 2: Find $\cos 2\theta$ given $\cos\theta = -\frac{1}{3}$

Example 3: Find the exact value of $\sin 15°$ using the half-angle formula

Verify: $\frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$

Verify: $\tan\theta + \cot\theta = \sec\theta\csc\theta$

Verify: $\frac{1 + \sin\theta}{\cos\theta} = \frac{\cos\theta}{1 - \sin\theta}$

Example 1: Simplify $\frac{\sin 2\theta}{1 + \cos 2\theta}$

Example 2: Find $\sin 75°\cos 15° + \cos 75°\sin 15°$

Example 3: Simplify $\frac{\sec(-\theta)}{\csc(90° - \theta)}$

Example 4: Verify $\frac{\sin 2\theta}{\sin\theta} - \frac{\cos 2\theta}{\cos\theta} = \sec\theta$