Verifying Additional Trigonometric Identities (College Board AP® Precalculus): Study Guide

Verifying additional trigonometric identities

What does it mean to verify a trigonometric identity?

• A trigonometric identity is

  • an equation involving trigonometric functions

  • that is true for all values of the variable in the common domain of both sides

• To verify an identity means to show that the two sides of the equation are equivalent

  • i.e. that one side can be rewritten to match the other

  • using known identities and algebraic manipulations

• Verifying an identity is not the same as solving an equation

  • You do not move terms from one side to the other

  • Instead, you transform one side step by step until it looks exactly like the other side

What tools are used to verify identities?

• Any identity or algebraic rule already known can be used, including

  • The Pythagorean identity and its rearranged forms

    • ${\sin }^2\theta +{\cos }^2\theta =1$, ${\sec }^2\theta -{\tan }^2\theta =1$, ${\csc }^2\theta -{\cot }^2\theta =1$

  • The sum, difference, and double-angle identities

  • The reciprocal relationships

    • $\sec \theta =\frac{1}{\cos \theta }$, $\csc \theta =\frac{1}{\sin \theta }$, $\cot \theta =\frac{1}{\tan \theta }$

  • The quotient identities

    • $\tan \theta =\frac{\sin \theta }{\cos \theta }$, $\cot \theta =\frac{\cos \theta }{\sin \theta }$

  • Standard algebraic manipulations

    • combining fractions, factoring, expanding, distributing, etc.

What is the general process for verifying an identity?

• Start by picking one side to work with

  • It is usually best to start with the more complex side

    • since simplifying is typically easier than complicating

• Look at the target side

  • I.e. the side you'll need to transform your 'starting side' into

  • What form does it take?

    • This often suggests which identities or manipulations will be useful

  • E.g. if the target side contains only $\sin$ and $\cos$

    • consider converting all the functions on the starting side into $\sin$ and $\cos$

  • Or if the target side contains a single trig function

    • look for a way to cancel or combine terms on the starting side

• Transform the starting side step by step, using known identities and algebraic rules

  • until it matches the target side exactly

• At each step, write clearly which identity or rule is being applied

What strategies are commonly useful?

• Convert to sine and cosine

  • Rewriting all trig functions on one side in terms of $\sin$ and $\cos$ is often a good way to start

    • particularly when the identity involves reciprocal functions

• Combine fractions

  • If one side has multiple fractions, combining them over a common denominator often reveals opportunities for further simplification

• Look for Pythagorean patterns

  • Expressions like $1-{\sin }^{2}x$, $1+{\tan }^{2}x$, or $(1-\cos x)(1+\cos x)$ can often be simplified using the Pythagorean identity

• Factor when possible

  • Factoring numerators or denominators can reveal cancellations

• Work from both sides if necessary

  • If it is not clear how to proceed from one side, you can work on both sides separately to get them into a common simpler form

  • But the final write-up should still show one side being transformed into the other