Overview

Identities are the backbone of trig simplification. Most contest problems use just a few core identities, applied quickly.

Pythagorean Identities

$\sin^2\theta + \cos^2\theta = 1.$

Divide by $\cos^2\theta$ or $\sin^2\theta$ to get:

$1 + \tan^2\theta = \sec^2\theta,\quad 1 + \cot^2\theta = \csc^2\theta.$

Even-Odd Identities

• $\sin(-\theta) = -\sin\theta$.
• $\cos(-\theta) = \cos\theta$.
• $\tan(-\theta) = -\tan\theta$.

Cofunction Identities

• $\sin(90^\circ-\theta) = \cos\theta$.
• $\cos(90^\circ-\theta) = \sin\theta$.
• $\tan(90^\circ-\theta) = \cot\theta$.

Core Skills

Pick a Target Identity

Decide whether to convert everything to $\sin$ and $\cos$ or use a Pythagorean identity to simplify the expression.

Track Quadrant Signs

When taking square roots, use the quadrant of $\theta$ to choose the sign.

Use Reciprocal Relations

Swap between $\tan$ and $\sin/\cos$ when it simplifies the expression.

Worked Example

If $\sin\theta=3/5$ and $\theta$ is in quadrant I, find $\cos\theta$ and $\tan\theta$.

$\cos\theta = \sqrt{1-9/25} = 4/5$, and $\tan\theta = (3/5)/(4/5)=3/4$.

More Examples

Example 1: Pythagorean Identity

If $\tan\theta=5/12$ in quadrant I, find $\sec\theta$.

$1+\tan^2\theta=\sec^2\theta$ gives $\sec\theta=13/12$.

Example 2: Even-Odd

Simplify $\sin(-x)\cos(-x)$.

$\sin(-x)\cos(-x) = (-\sin x)(\cos x) = -\sin x\cos x$.

Example 3: Cofunction

Simplify $\cos(90^\circ-\theta)+\sin(90^\circ-\theta)$.

$\sin\theta+\cos\theta$.

Strategy Checklist

• Decide whether to convert to $\sin$ and $\cos$ first.
• Use Pythagorean identities to eliminate squares.
• Track the correct sign with quadrant info.

Common Pitfalls

• Dropping the $\pm$ when taking square roots.
• Using cofunction identities without adjusting the angle to $90^\circ-\theta$.
• Mixing degrees and radians in angle arguments.

Practice Problems