click to advance
Unit 5 · Analytic Trigonometry
5.2 Verifying
Trigonometric Identities
Objective: Verify trigonometric identities
Work one side only Pythagorean Identities Common denominator Multiply by conjugate Convert to sin & cos Factor
Launch
Strategy Sort — Best First Move

At your boards: Sort each identity by the best first move. Choose the strategy that gets the proof started cleanly, but do not finish the algebra yet.

Touchscreen fallback: Tap a card, then tap a bucket. On laptops, you can drag cards directly and they will snap into the grid.

Card Bank
Pythagorean Identity
Common Denominator
Convert to sin and cos
Multiply by Conjugate
Factor
Teacher key: A → Pythagorean identity, B → common denominator, C → convert to sin and cos, D → multiply by conjugate, E → factor, F → factor.
Concept
What is an identity?

An identity is an equation that is true for all values in the domain of the variable.

Verifying (also called "proving") an identity means showing that one side can be algebraically transformed into the other — using known identities and valid algebra steps.

The cardinal rule: You work on one side at a time. You cannot cross the equal sign — do not add to both sides, multiply both sides, or square both sides. Each step transforms the same side.

Tip: When in doubt, start with the more complicated side, or convert everything to sine and cosine — then look for a path to the other side.

Strategy
Guidelines for Verifying
💡 Tip: Try something! Even an attempt that leads to a dead end gives insight into what will work.

Do

  • Work with the more complicated side
  • Try a common denominator
  • Factor expressions
  • Convert all to sine and cosine
  • Multiply by the conjugate
  • Apply Pythagorean Identities
  • Work each side separately toward a common middle

Do NOT

  • Cross the equal sign
  • Multiply both sides by anything
  • Add to both sides
  • Square both sides
Example 1
Verify the identity
\[\dfrac{\sec^2\theta - 1}{\sec^2\theta} = \sin^2\theta\]
Strategy: Use a Pythagorean identity, then convert to sine and cosine.
Left side
\(\dfrac{\sec^2\theta - 1}{\sec^2\theta}\)
\(= \dfrac{\tan^2\theta}{\sec^2\theta}\) Pythagorean identity: \(\sec^2\theta - 1 = \tan^2\theta\)
\(= \tan^2\theta \cdot \cos^2\theta\) since \(\dfrac{1}{\sec^2\theta} = \cos^2\theta\)
\(= \dfrac{\sin^2\theta}{\cos^2\theta} \cdot \cos^2\theta\) substitute \(\tan^2\theta = \dfrac{\sin^2\theta}{\cos^2\theta}\)
\(= \sin^2\theta \checkmark\) \(\cos^2\theta\) cancels
Another Way
Example 1 — Split the fraction
\[\dfrac{\sec^2\theta - 1}{\sec^2\theta} = \sin^2\theta\]
Strategy: Split the numerator into two separate fractions, then simplify each term.
Left side
\(\dfrac{\sec^2\theta - 1}{\sec^2\theta}\)
\(= \dfrac{\sec^2\theta}{\sec^2\theta} - \dfrac{1}{\sec^2\theta}\) split into two fractions with the same denominator
\(= 1 - \cos^2\theta\) \(\dfrac{\sec^2\theta}{\sec^2\theta} = 1\) and \(\dfrac{1}{\sec^2\theta} = \cos^2\theta\)
\(= \sin^2\theta \checkmark\) Pythagorean identity: \(1 - \cos^2\theta = \sin^2\theta\)
Example 2
Verify the identity
\[\dfrac{1}{1 - \sin\alpha} + \dfrac{1}{1 + \sin\alpha} = 2\sec^2\alpha\]
Strategy: Get a common denominator and add the fractions.
Left side
\(\dfrac{1}{1 - \sin\alpha} + \dfrac{1}{1 + \sin\alpha}\)
\(= \dfrac{1 + \sin\alpha}{(1 - \sin\alpha)(1 + \sin\alpha)} + \dfrac{1 - \sin\alpha}{(1 - \sin\alpha)(1 + \sin\alpha)}\) multiply each fraction to build a common denominator
\(= \dfrac{(1 + \sin\alpha) + (1 - \sin\alpha)}{1 - \sin^2\alpha}\) add numerators; denominator is a difference of squares: \((1-\sin\alpha)(1+\sin\alpha) = 1 - \sin^2\alpha\)
\(= \dfrac{2}{\cos^2\alpha}\) numerator: \(1 + \sin\alpha + 1 - \sin\alpha = 2\); denominator: \(1 - \sin^2\alpha = \cos^2\alpha\)
\(= 2\sec^2\alpha \checkmark\) since \(\dfrac{1}{\cos^2\alpha} = \sec^2\alpha\)
Example 3
Verify the identity
\[\tan x + \cot x = \sec x \csc x\]
Strategy: Convert the left side to sine and cosine, then add the fractions.
Left side
\(\tan x + \cot x\)
\(= \dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x}\) convert using \(\tan x = \dfrac{\sin x}{\cos x}\) and \(\cot x = \dfrac{\cos x}{\sin x}\)
\(= \dfrac{\sin^2 x + \cos^2 x}{\cos x \cdot \sin x}\) add with common denominator \(\cos x \sin x\)
\(= \dfrac{1}{\cos x \cdot \sin x}\) Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\)
\(= \sec x \csc x \checkmark\) split: \(\dfrac{1}{\cos x} \cdot \dfrac{1}{\sin x} = \sec x \csc x\)
Example 4
Verify the identity
\[\sec x + \tan x = \dfrac{\cos x}{1 - \sin x}\]
Strategy: Work with the right side — multiply by the conjugate of the denominator.
Right side
\(\dfrac{\cos x}{1 - \sin x}\)
\(= \dfrac{\cos x}{1 - \sin x} \cdot \dfrac{1 + \sin x}{1 + \sin x}\) multiply by the conjugate of the denominator
\(= \dfrac{\cos x(1 + \sin x)}{1 - \sin^2 x}\) denominator: \((1 - \sin x)(1 + \sin x) = 1 - \sin^2 x\)
\(= \dfrac{\cos x(1 + \sin x)}{\cos^2 x} = \dfrac{1 + \sin x}{\cos x}\) replace \(1 - \sin^2 x = \cos^2 x\), then cancel one factor of \(\cos x\)
\(= \dfrac{1}{\cos x} + \dfrac{\sin x}{\cos x} = \sec x + \tan x \checkmark\) split the fraction into two terms
Example 5
Verify the identity
\[\dfrac{\cot^2\theta}{1 + \csc\theta} = \dfrac{1 - \sin\theta}{\sin\theta}\]
Strategy: Substitute a Pythagorean identity in the numerator, then factor as a difference of squares.
Left side
\(\dfrac{\cot^2\theta}{1 + \csc\theta}\)
\(= \dfrac{\csc^2\theta - 1}{1 + \csc\theta}\) Pythagorean identity: \(\cot^2\theta = \csc^2\theta - 1\)
\(= \dfrac{(\csc\theta - 1)(\csc\theta + 1)}{1 + \csc\theta}\) factor numerator as a difference of squares
\(= \csc\theta - 1\) cancel \((\csc\theta + 1)\) from numerator and denominator
\(= \dfrac{1}{\sin\theta} - \dfrac{\sin\theta}{\sin\theta}\) replace \(\csc\theta = \dfrac{1}{\sin\theta}\) and rewrite \(1 = \dfrac{\sin\theta}{\sin\theta}\)
\(= \dfrac{1 - \sin\theta}{\sin\theta} \checkmark\)
Another Way
Example 5 — Multiply by conjugate
\[\dfrac{\cot^2\theta}{1 + \csc\theta} = \dfrac{1 - \sin\theta}{\sin\theta}\]
Strategy: Multiply numerator and denominator by the conjugate \((1 - \csc\theta)\).
Left side
\(\dfrac{\cot^2\theta}{1 + \csc\theta}\)
\(= \dfrac{\cot^2\theta}{1 + \csc\theta} \cdot \dfrac{1 - \csc\theta}{1 - \csc\theta}\) multiply by the conjugate of the denominator
\(= \dfrac{\cot^2\theta\,(1 - \csc\theta)}{1 - \csc^2\theta}\) denominator: \((1 + \csc\theta)(1 - \csc\theta) = 1 - \csc^2\theta\)
\(= \dfrac{\cot^2\theta\,(1 - \csc\theta)}{-\cot^2\theta} = -(1 - \csc\theta)\) since \(1 - \csc^2\theta = -\cot^2\theta\); cancel \(\cot^2\theta\)
\(= \csc\theta - 1 = \dfrac{1}{\sin\theta} - 1 = \dfrac{1 - \sin\theta}{\sin\theta} \checkmark\) distribute the negative, then convert \(\csc\theta\) and combine over \(\sin\theta\)
Example 6
Verify the identity
\[\sin^3 x \cos^4 x = (\cos^4 x - \cos^6 x)\sin x\]
Strategy: Work with the right side — factor, then apply a Pythagorean identity.
Right side
\((\cos^4 x - \cos^6 x)\sin x\)
\(= \cos^4 x(1 - \cos^2 x)\sin x\) factor out \(\cos^4 x\)
\(= \cos^4 x \cdot \sin^2 x \cdot \sin x\) Pythagorean identity: \(1 - \cos^2 x = \sin^2 x\)
\(= \sin^3 x \cos^4 x \checkmark\) combine: \(\sin^2 x \cdot \sin x = \sin^3 x\)
Another Way
Example 6 — Work from the left side
\[\sin^3 x \cos^4 x = (\cos^4 x - \cos^6 x)\sin x\]
Strategy: Rewrite \(\sin^3 x\) as \(\sin^2 x \cdot \sin x\), then substitute \(\sin^2 x = 1 - \cos^2 x\).
Left side
\(\sin^3 x \cos^4 x\)
\(= \sin^2 x \cdot \sin x \cdot \cos^4 x\) rewrite \(\sin^3 x = \sin^2 x \cdot \sin x\)
\(= (1 - \cos^2 x) \cdot \sin x \cdot \cos^4 x\) Pythagorean identity: \(\sin^2 x = 1 - \cos^2 x\)
\(= (\cos^4 x - \cos^6 x)\sin x \checkmark\) distribute \(\sin x \cdot \cos^4 x\): \(\cos^4 x \sin x - \cos^6 x \sin x\)