Trigonometric Simplification Quiz: Simplifying Trigonometric Expressions

Use the Pythagorean identity sin^2θ + cos^2θ = 1. No matter the value of θ, the sum of the squares of sine and cosine equals 1.

From 1 + tan^2θ = sec^2θ, rearrange to sec^2θ − tan^2θ = 1.

From 1 + cot^2θ = csc^2θ, subtract 1 to get csc^2θ − 1 = cot^2θ.

Since 1 + tan^2θ = sec^2θ, the expression becomes sec^2θ/sec^2θ = 1.

Since 1 + cot^2θ = csc^2θ, the expression equals csc^2θ − csc^2θ = 0.

sin^2θ = 1 − cos^2θ by sin^2θ + cos^2θ = 1. Also, 1/csc^2θ = sin^2θ since csc^2θ = 1/sin^2θ. The others simplify to 1 or are not identities for sin^2θ.

cot^2θ = csc^2θ − 1 and cot^2θ = cos^2θ/sin^2θ = 1/tan^2θ. (1 − cos^2θ)/sin^2θ = sin^2θ/sin^2θ = 1. sec^2θ − 1 = tan^2θ.

1 − sin^2θ = cos^2θ. Therefore (1 − sin^2θ)/cos^2θ = cos^2θ/cos^2θ = 1.

From sin^2θ + cos^2θ = 1, subtract sin^2θ to get 1 − sin^2θ = cos^2θ.

The identity is 1 + tan^2θ = sec^2θ. Therefore sec^2θ = 1 + tan^2θ, not 1 − tan^2θ.

By the Pythagorean identities: sin^2θ + cos^2θ = 1, sec^2θ − tan^2θ = 1, csc^2θ − cot^2θ = 1. Also, (1 − sin^2θ)/cos^2θ = cos^2θ/cos^2θ = 1. tan^2θ − sec^2θ = −1.

From 1 + cot^2θ = csc^2θ, rearrange to csc^2θ − cot^2θ = 1.

sec^2θ − 1 = tan^2θ. So (sec^2θ − 1)/tan^2θ = tan^2θ/tan^2θ = 1.

tan^2θ + 1 = sec^2θ. csc^2θ relates to cot^2θ by 1 + cot^2θ = csc^2θ.

1 − cos^2θ = sin^2θ. Thus sin^2θ/(1 − cos^2θ) = sin^2θ/sin^2θ = 1.

1 − sin^2θ = cos^2θ and 1 − cos^2θ = sin^2θ. Therefore (1 − sin^2θ)/(1 − cos^2θ) = cos^2θ/sin^2θ = cot^2θ.

sec^2θ = 1 + tan^2θ and sec^2θ = 1/cos^2θ. csc^2θ − cot^2θ = 1. 1 − tan^2θ is not an identity. (1 + sin^2θ)/cos^2θ = sec^2θ + tan^2θ, not just sec^2θ.

sec^2θ − 1 = tan^2θ and 1 + tan^2θ = sec^2θ. So the fraction is tan^2θ/sec^2θ = (sin^2θ/cos^2θ)/(1/cos^2θ) = sin^2θ.

From 1 + tan^2θ = sec^2θ, rearrange to tan^2θ = sec^2θ − 1.

Since sin^2θ + cos^2θ = 1, then 1 − cos^2θ = sin^2θ.