Trigonometric 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

Useful Rearrangements

📝 Worked Examples

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

Replace $1 - \cos^2\theta$ with $\sin^2\theta$:

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

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

🎯 Simplification Strategy

Decision Tree for Pythagorean Identities

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.

Concept Check 🎯

Pythagorean Identity Practice 🧮

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

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)

Identity Matching 🔽

Exit Quiz ✅

🔄 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

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

Quotient Identities

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

Why Convert to Sine & Cosine?

📝 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

Common Products That Equal 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 🔽

Exit Quiz ✅

🪞 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

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 $(\cos\theta,\,-\sin\theta)$.

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 $x$ and $y$ swap!

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

Quick Decision Table

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 🔽

Exit Quiz ✅

➕ 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 $\sin A$, $\cos A$, $\sin B$, $\cos B$.

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

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

When to Use Sum/Difference Formulas

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., $\sin 30° = 1/2$)

Formula Matching 🔽

Exit Quiz ✅

✖️ 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 $\sin\frac{\theta}{2}$, etc., in terms of $\cos\theta$.

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:

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 $\cos 30° = \frac{\sqrt{3}}{2}$.

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

($+$ 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

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., $\cos 2\theta = 1 - 2(3/5)^2 = 1 - 18/25 = 7/25$)

3) If $\cos\theta = \frac{3}{5}$ and $\sin\theta = \frac{4}{5}$, find $\tan 2\theta$. Write as a fraction. (e.g., with $\tan\theta = 1$, $\tan 2\theta = \frac{2(1)}{1-1^2}$ is undefined)

Formula Recognition 🔽

Exit Quiz ✅

✅ 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

📝 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?

Common Mistakes to Avoid

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 $\frac{1-\cos\theta}{1-\cos\theta}$. The new denominator $1 - \cos^2\theta$ equals $\sin^{\,n}\theta$. What is $n$? (e.g., $1 - a^2$ might become $b^3$, so $n = 3$)

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 $1-\cos^2\theta$ becomes $\sin^{\,k}\theta$. What is $k$? (e.g., $1 - b^2$ might yield $c^4$, so $k = 4$)

Strategy Matching 🔽

Exit Quiz ✅

🧩 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

🗺️ Identity Selection Flowchart

What Do I See? → What Do I Use?

Multi-Step Strategy

1. Scan — Identify the identity types present
2. Convert — Rewrite everything in sin/cos if mixed functions appear
3. Combine — Get a single fraction if multiple terms
4. Substitute — Apply Pythagorean, double-angle, etc.
5. 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 $\cos 2\theta$)

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 ✅