### Home > PC > Chapter 4 > Lesson 4.2.4 > Problem4-112

4-112.

Verify the following trigonometric identities. Intermediate steps may involve complex fractions.

1. $\operatorname{cot}^2 x(\operatorname{sec}^2 x − 1) = 1$

Substitute a Pythagorean Identity for $\operatorname{sec}^2x$.

1. $\operatorname{sin}^2 α − \operatorname{cos}^2 α = 1 − 2 \operatorname{cos}^2 α$

Substitute the Fundamental Pythagorean Identity into the left side of the equation for $\operatorname{sin}^2α$.

1. $\operatorname{tan}^2 β + 6 = \operatorname{sec}^2 β + 5$

Substitute a Pythagorean Identity for $\operatorname{sec}^2x$.

1. $\text{cos }y + \text{sin }y · \text{tan }y = \text{sec }y$

Rewrite $\text{tan }y$ as $\text{sin }y$ / $\text{cos }y$.
Get common denominators and simplify.