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Pre-Calculus Trigonometric identities

Simplifying trig expressions using identities

20 practice questions 0 video lessons Theory + worked examples

Simplifying and Verifying Trig Expressions

Texas Precalculus (TEKS) • Standard P.5(M) • Trigonometric Identities

Simplifying and Verifying Trig Expressions is a topic in Trigonometric Identities in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.5(M), which requires students to simplify trigonometric expressions using identities.

Verifying a trigonometric identity transforms one side, usually by converting to sine and cosine and applying the Pythagorean identity, until it matches the other side.

Texas Precalculus (TEKS) › Trigonometric Identities › Simplifying and Verifying Trig Expressions  —  Standard P.5(M)

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Theory

To verify an identity is to show two expressions are equal for all valid angles. The reliable strategy:

  1. Work on one side only — usually the more complicated one.
  2. Convert everything to \(\sin\) and \(\cos\).
  3. Use the fundamental identities (especially \(\sin^2+\cos^2=1\)) and combine fractions.
  4. Simplify until it matches the other side.
Transform, don't “solve.” Never move terms across the equals sign as if it were an equation — that assumes what you're trying to prove.
Strategy for verifying a trig identity Work with the more complicated side, convert everything to sine and cosine, and transform it until it matches the other side. complicated side→sin & cos+ Pythagoreansimplify until it equals the other sideLHS = RHS ✓
Verification strategy: start on the complicated side, convert to \(\sin/\cos\), and simplify to the other side.
Handy simplifications Handy simplifications Handy simplifications 1 − cos²x = sin²x sec x cos x = 1 tan x cos x = sin x
Frequently used simplifications.

The workhorse identities for simplifying:

\[\sin^2 x+\cos^2 x=1,\quad \tan x=\dfrac{\sin x}{\cos x},\quad \sec x=\dfrac{1}{\cos x}\]
sine squared plus cosine squared equals one; tangent is sine over cosine; secant is one over cosine
Rewriting in \(\sin/\cos\) almost always exposes the cancellation you need.

How to verify a trig identity

  1. Choose the busier side to work on.
  2. Convert all functions to sine and cosine.
  3. Combine fractions over a common denominator and apply \(\sin^2+\cos^2=1\).
  4. Simplify step by step until the side becomes the other one.
Example 1 — Simplify to a single function
Simplify \(\dfrac{1-\cos^2 x}{\sin x}\).
Solution

Replace \(1-\cos^2 x\) with \(\sin^2 x\) (Pythagorean identity), then cancel.

\(\dfrac{1-\cos^2 x}{\sin x}\)\(=\)\(\dfrac{\sin^2 x}{\sin x}\)
\(=\)\(\sin x\)
expression simplifies to sine x
Example 2 — Verify an identity
Verify \(\tan x\cos x=\sin x\).
Solution

Work the left side; rewrite \(\tan x\) as \(\dfrac{\sin x}{\cos x}\).

\(\tan x\cos x\)\(=\)\(\dfrac{\sin x}{\cos x}\cdot\cos x\)
\(=\)\(\sin x\)

Left side equals right side, so the identity holds.

left side reduces to sine x, matching the right side
Example 3 — Combine fractions
Verify \(\dfrac{1}{\sin x}-\dfrac{\cos^2 x}{\sin x}=\sin x\).
Solution

Same denominator, so combine numerators, then use the Pythagorean identity.

\(=\)\(\dfrac{1-\cos^2 x}{\sin x}\)
\(=\)\(\dfrac{\sin^2 x}{\sin x}=\sin x\)
combining the fractions and simplifying gives sine x
Example 4 — A secant identity
Verify \(\sec^2 x-1=\tan^2 x\).
Solution

This is a rearrangement of the Pythagorean identity \(1+\tan^2 x=\sec^2 x\).

\(\sec^2 x-1\)\(=\)\((1+\tan^2 x)-1\)
\(=\)\(\tan^2 x\)
secant squared minus one equals tangent squared

Common pitfalls

Don't cross the equals sign. Transform one side; don't manipulate both as if solving an equation.
Convert to \(\sin/\cos\) early. It's the most dependable route through a stubborn expression.
Watch for a hidden Pythagorean identity. \(1-\cos^2 x\), \(\sec^2 x-1\), and similar forms simplify instantly.

Frequently asked questions

How do you verify a trig identity?

Work one side only, convert everything to sine and cosine, apply the fundamental identities, and simplify until it equals the other side.

Why shouldn't you move terms across the equals sign?

Because that assumes the identity is already true. A valid proof transforms one side independently until it matches the other.

What is the best first step?

Rewrite every function in terms of sine and cosine. It usually reveals the cancellation or common denominator you need.

Which identity is used most often?

The Pythagorean identity \(\sin^2 x+\cos^2 x=1\), along with its rearranged forms like \(1-\cos^2 x=\sin^2 x\).

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