Home / Courses / Precalculus / Unit 9

Precalculus Unit 9: Trig identities

Trig identities are equations true for every angle. They let you rewrite expressions to simplify or solve them.

Covers: Pythagorean, sum/difference, double-angle identities

Study guide

Trig identities are equations true for every angle where both sides are defined. They let you rewrite expressions, simplify, and solve equations.

The Pythagorean identities come from the unit circle: sin²θ + cos²θ = 1, and dividing by cos²θ or sin²θ gives 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. The reciprocal identities define sec θ = 1/cos θ, csc θ = 1/sin θ and cot θ = 1/tan θ, and the quotient identity is tan θ = sin θ/cos θ.

The sum and difference identities expand trig functions of combined angles: sin(A ± B) = sin A cos B ± cos A sin B, and cos(A ± B) = cos A cos B ∓ sin A sin B (note the sign flip). They let you find exact values like sin 75° = sin(45° + 30°).

Setting A = B gives the double-angle identities: sin 2θ = 2 sin θ cos θ, and cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1. Rearranging the cosine forms gives the power-reducing formulas, sin²θ = (1 − cos 2θ)/2 and cos²θ = (1 + cos 2θ)/2.

To prove an identity, work on one side only (usually the more complicated one), rewrite in terms of sine and cosine, combine fractions, factor, and use known identities until it matches the other side.

sin(θ/2) = ±√((1 − cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2), with the sign chosen by the quadrant of θ/2. These follow from the power-reducing formulas with θ replaced by θ/2. tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A·tan B). For example, tan 15° = tan(45° − 30°).

Key ideas

• The Pythagorean identity comes straight from the unit circle.
• Reciprocal identities: csc = 1/sin, sec = 1/cos, cot = 1/tan.
• Sum and difference formulas break angles apart.
• Double-angle formulas come from setting both angles equal in the sum formulas.
• To verify an identity, work on one side only.

Formulas and rules

Common mistakes

• Writing sin(A + B) = sin A + sin B.
• Moving terms across the equals sign while verifying an identity.

Words to know

Pythagorean identity

sin²θ + cos²θ = 1

sin 2θ

2 sin θ cos θ

cos(A + B)

cos A cos B − sin A sin B

1 + tan²θ

sec²θ

Three forms of cos 2θ

cos²θ − sin²θ, 1 − 2sin²θ, 2cos²θ − 1

Reciprocal identities

sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ

Quotient identity

tan θ = sin θ/cos θ

1 + cot²θ

csc²θ

sin(A + B)

sin A cos B + cos A sin B

Power-reducing formula

sin²θ = (1 − cos 2θ)/2

Proving an identity

Work on one side, rewrite in sine and cosine, simplify until both sides match.

Sample practice questions

13 more questions like these, from level 1 to 5, are in the practice set.