Question

Considering only the values of θ for which sec θ (1-sin²θ ) is defined, which of the following expressions is equivalent to sec θ (1-sin²θ ) ?

A. Cos θ
B. sec³θ
C. csc θ
D. sin θ tan θ
E. sec²θ

Answer 1

The expression equivalent to sec θ (1-sin²θ) is 1/sec θ, which is the same as cos θ.

Step-by-step explanation:

To determine the expression that is equivalent to sec θ (1-sin²θ), we need to simplify the expression. Let's start by expanding the expression using trigonometric identities. Recall that sin²θ + cos²θ = 1. Substituting this into the expression gives us sec θ (1-sin²θ) = sec θ (cos²θ). Next, we can use the identity cos²θ = 1/sec²θ to simplify further. Therefore, the equivalent expression is 1/sec θ, which is the same as cos θ (option A).