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Mom's Basement University

Verifying Trigonometric Identities

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn strategies for verifying (proving) trigonometric identities by transforming one side of the equation into the other.

The Concept

Section titled “The Concept”

To verify a trigonometric identity, you must show that the equation is true for all values of the variable (where defined). You do this by simplifying one or both sides until they match.

Key strategies:

  • Start with the more complicated side and try to transform it into the simpler side
  • Use fundamental identities (reciprocal, quotient, Pythagorean)
  • Factor, multiply by conjugates, or write everything in terms of sine and cosine when stuck
  • Work on only one side at a time. Never move terms from one side to the other
  • If needed, work on both sides separately until they meet in the middle

Common techniques:

  • Replace tan with sin/cos, sec with 1/cos, etc.
  • Use Pythagorean identities to replace sin² or cos²
  • Multiply numerator and denominator by a conjugate to rationalize

Worked Example

Section titled “Worked Example”

Verify: sin⁡θ1−cos⁡θ=1+cos⁡θsin⁡θ\frac{\sin\theta}{1 - \cos\theta} = \frac{1 + \cos\theta}{\sin\theta}

Start with the left side. Multiply numerator and denominator by the conjugate (1 + cos θ):

sin⁡θ1−cos⁡θ⋅1+cos⁡θ1+cos⁡θ\frac{\sin\theta}{1 - \cos\theta} \cdot \frac{1 + \cos\theta}{1 + \cos\theta}

The denominator becomes a difference of squares:

=sin⁡θ(1+cos⁡θ)1−cos⁡2θ= \frac{\sin\theta(1 + \cos\theta)}{1 - \cos^2\theta}

By the Pythagorean identity, 1 - cos²θ = sin²θ

=sin⁡θ(1+cos⁡θ)sin⁡2θ= \frac{\sin\theta(1 + \cos\theta)}{\sin^2\theta}

Cancel one sin θ from top and bottom:

=1+cos⁡θsin⁡θ= \frac{1 + \cos\theta}{\sin\theta}

This matches the right side. Identity verified.

Verify: tan⁡2θ+1=sec⁡2θ\tan^2\theta + 1 = \sec^2\theta

Start with the left side. Replace tan with sin/cos:

tan⁡2θ+1=sin⁡2θcos⁡2θ+1\tan^2\theta + 1 = \frac{\sin^2\theta}{\cos^2\theta} + 1

Write 1 as cos²θ / cos²θ so we have a common denominator:

=sin⁡2θcos⁡2θ+cos⁡2θcos⁡2θ=sin⁡2θ+cos⁡2θcos⁡2θ= \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{\sin^2\theta + \cos^2\theta}{\cos^2\theta}

By the Pythagorean identity, the numerator equals 1:

=1cos⁡2θ=sec⁡2θ= \frac{1}{\cos^2\theta} = \sec^2\theta

This matches the right side. Identity verified.

Real-World Application

Section titled “Real-World Application”

Verifying identities is essential in:

  • Simplifying complex waveforms in audio and signal processing
  • Solving trigonometric equations in physics and engineering
  • Proving relationships in electrical circuit analysis
  • Optimizing formulas in computer graphics and animation

Example: engineers use identities to simplify expressions for alternating current and voltage phase relationships. A complicated expression can often be reduced to something much simpler using the right identity.

Quiz

Section titled “Quiz”
When verifying an identity, you should:
A good strategy when stuck is to:
To verify $\tan\theta = \frac{\sin\theta}{\cos\theta}$, you would:
Multiplying by a conjugate is useful when:
The final goal when verifying an identity is to show: