They are powerful tools for simplifying expressions and solving equations.

Reciprocal Identities

$\csc(\theta) = \frac{1}{\sin(\theta)}$

$\sec(\theta) = \frac{1}{\cos(\theta)}$

$\cot(\theta) = \frac{1}{\tan(\theta)}$

Quotient Identities

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Pythagorean Identities

These come from $x^2 + y^2 = 1$ on the unit circle:

Primary Form

$\sin^2(\theta) + \cos^2(\theta) = 1$

Divide by $\cos^2(\theta)$

$\tan^2(\theta) + 1 = \sec^2(\theta)$

Divide by $\sin^2(\theta)$

$1 + \cot^2(\theta) = \csc^2(\theta)$

Even-Odd Identities

Even functions (symmetric about y-axis): $\cos(-\theta) = \cos(\theta)$ $\sec(-\theta) = \sec(\theta)$

Odd functions (symmetric about origin): $\sin(-\theta) = -\sin(\theta)$ $\tan(-\theta) = -\tan(\theta)$ $\csc(-\theta) = -\csc(\theta)$ $\cot(-\theta) = -\cot(\theta)$

Cofunction Identities

Cofunctions of complementary angles are equal:

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

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

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

How to Use Identities

1. Simplify expressions: Replace complex trig expressions with simpler ones
2. Prove other identities: Use known identities to verify new ones
3. Solve equations: Transform equations into solvable forms
4. Evaluate expressions: Find exact values

Strategy for Proving Identities

1. Start with the more complicated side
2. Use fundamental identities to rewrite terms
3. Look for opportunities to factor or combine fractions
4. Convert everything to sines and cosines if stuck
5. Never move terms from one side to the other (work on each side independently)

Common Mistakes to Avoid

❌ Don't treat identities like equations and cross-multiply ❌ Don't forget to square correctly: $(\sin \theta)^2 = \sin^2 \theta$ ❌ Don't cancel terms that aren't factors

So our expression becomes: $\tan(\theta) + \cot(\theta)$

Step 2: Find a common denominator. $= \frac{\sin(\theta)}{\cos(\theta)} + \frac{\cos(\theta)}{\sin(\theta)}$

$= \frac{\sin^2(\theta) + \cos^2(\theta)}{\sin(\theta)\cos(\theta)}$

Step 3: Use the Pythagorean identity. $\sin^2(\theta) + \cos^2(\theta) = 1$

$= \frac{1}{\sin(\theta)\cos(\theta)}$

Step 4: Write in terms of reciprocal identities. $= \frac{1}{\sin(\theta)} \cdot \frac{1}{\cos(\theta)} = \csc(\theta)\sec(\theta)$

Answer: $\csc(\theta)\sec(\theta)$ or $\frac{1}{\sin(\theta)\cos(\theta)}$