Trigonometric Identities

1 + tan²θ, 1/cos²θ, cosec²θ - cot²θ

From the identity 1 + tan²θ = sec²θ. Also, sec θ = 1/cos θ, so sec²θ = 1/cos²θ. Using cosec²θ = 1 + cot²θ and sec²θ cosec²θ = 1, we get cosec²θ - cot²θ = sec²θ.

1 + cot²θ = cosec²θ

This is one of the three fundamental trigonometric identities. The other two are sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ.

5/12

First find cos θ using sin²θ + cos²θ = 1. cos²θ = 1 - (5/13)² = 1 - 25/169 = 144/169. So cos θ = 12/13. Then tan θ = sin θ/cos θ = (5/13)/(12/13) = 5/12.

tan²θ

From the identity 1 + tan²θ = sec²θ, we can rearrange to get sec²θ - 1 = tan²θ.