Precalculus: Trigonometric Identities Worksheet

Question 1

Simplify the expression sin^2 x + cos^2 x + tan^2 x.

Accepted Answer

The expression simplifies to sec^2 x because sin^2 x + cos^2 x = 1, so the expression becomes 1 + tan^2 x = sec^2 x.

Question 2

Verify the identity (1 - cos^2 θ) / sin θ = sin θ.

Accepted Answer

Using 1 - cos^2 θ = sin^2 θ, the left side becomes sin^2 θ / sin θ. For sin θ not equal to 0, this simplifies to sin θ, so the identity is verified where both sides are defined.

Question 3

Simplify the expression (sec x - cos x) / tan x.

Accepted Answer

The expression simplifies to sin x. Rewriting in sine and cosine gives (1/cos x - cos x) / (sin x/cos x), which becomes ((1 - cos^2 x)/cos x)(cos x/sin x) = sin^2 x/sin x = sin x.

Question 4

Prove the identity tan θ + cot θ = sec θ csc θ.

Accepted Answer

The left side becomes sin θ/cos θ + cos θ/sin θ. With a common denominator, this is (sin^2 θ + cos^2 θ)/(sin θ cos θ). Since sin^2 θ + cos^2 θ = 1, the expression becomes 1/(sin θ cos θ) = sec θ csc θ.

Question 5

Rewrite cos^2 x using a double-angle identity.

Accepted Answer

The expression cos^2 x can be rewritten as (1 + cos 2x) / 2 using the identity cos 2x = 2cos^2 x - 1.

Question 6

Use an identity to find the exact value of sin 75°.

Accepted Answer

Using sin(45° + 30°), sin 75° = sin 45° cos 30° + cos 45° sin 30°. This equals (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Question 7

Use an identity to find the exact value of cos 15°.

Accepted Answer

Using cos(45° - 30°), cos 15° = cos 45° cos 30° + sin 45° sin 30°. This equals (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Question 8

Verify the identity sec^2 x - tan^2 x = 1.

Accepted Answer

The identity is verified because the Pythagorean identity 1 + tan^2 x = sec^2 x can be rearranged by subtracting tan^2 x from both sides, giving sec^2 x - tan^2 x = 1.

Question 9

Simplify the expression sin x csc x + cos x sec x.

Accepted Answer

The expression simplifies to 2 because sin x csc x = 1 and cos x sec x = 1 wherever the expression is defined.

Question 10

Rewrite tan(π/2 - x) using a cofunction identity.

Accepted Answer

The expression tan(π/2 - x) is equal to cot x by the cofunction identity for tangent.

Question 11

Prove the identity (1 + sin x)(1 - sin x) = cos^2 x.

Accepted Answer

The left side is a difference of squares: (1 + sin x)(1 - sin x) = 1 - sin^2 x. Since 1 - sin^2 x = cos^2 x, the identity is verified.

Question 12

Simplify the expression cos 2x + 2sin^2 x.

Accepted Answer

The expression simplifies to 1. Using cos 2x = 1 - 2sin^2 x, the expression becomes 1 - 2sin^2 x + 2sin^2 x = 1.

Question 13

If sin x = 3/5 and x is in Quadrant I, find cos 2x.

Accepted Answer

Since x is in Quadrant I, cos x = 4/5. Then cos 2x = cos^2 x - sin^2 x = (4/5)^2 - (3/5)^2 = 16/25 - 9/25 = 7/25.

Question 14

Solve the equation 2sin x cos x = 1 for 0 ≤ x < 2π.

Accepted Answer

Using the identity 2sin x cos x = sin 2x, the equation becomes sin 2x = 1. Thus 2x = π/2 or 5π/2 within the needed range, so x = π/4 or 5π/4.

Question 15

Use a half-angle identity to find the exact value of sin 15°.

Accepted Answer

Using sin(15°) = sin(30°/2), sin 15° = √((1 - cos 30°)/2). Since cos 30° = √3/2, this becomes √((1 - √3/2)/2) = √((2 - √3)/4) = √(2 - √3)/2.

Precalculus: Trigonometric Identities

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