3.10 Equivalent Representations of Trigonometric Functions

Sum and Difference Identities

Trig functions of sums/differences can be rewritten using sum and difference identities: sin(A±B) = sinA cosB ± cosA sinB, and cos(A±B) = cosA cosB ∓ sinA sinB. These let you rewrite expressions, evaluate exact values at non-standard angles, and prove other identities are equivalent representations of the same function.

Double-Angle Identities

Double-angle identities rewrite trig functions of 2θ in terms of θ: sin(2θ) = 2sinθ cosθ; cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. These come directly from the sum identities (A=B=θ) and are useful for simplifying expressions and solving equations.

Rewriting to Reveal Structure

The same trig expression can be written in multiple equivalent forms — useful for graphing, solving equations, or analyzing end behavior. Using Pythagorean identities (sin²θ+cos²θ=1) or algebraic manipulation lets you convert between sin, cos, and other forms to match a needed context.

Key terms

sin(A + B)

sinA cosB + cosA sinB

cos(A − B)

cosA cosB + sinA sinB

sin(2θ)

2sinθ cosθ

cos(2θ) — three equivalent forms

cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

Why rewrite trig expressions?

To simplify, solve equations, or reveal structure while representing the same function value in a more useful form.

Try an AP-style question

Which identity is true?

1. 1 + tan² θ = sec² θ
2. 1 − tan² θ = sec² θ
3. tan² θ − 1 = sec² θ
4. 1 + sec² θ = tan² θ