Question

let α and β be complementary angles. what is true about cos α and sin β? a. - cos α = sin β b. cos(-α)= - sin β c. cos α = - sin β d. cos α = sin β prove that sec(π/2 - θ)= csc θ.

Explanation:

Step1: Recall complementary - angle identity

If $\alpha$ and $\beta$ are complementary angles, then $\alpha+\beta = \frac{\pi}{2}$, so $\alpha=\frac{\pi}{2}-\beta$.

Step2: Use cosine - sine relationship

We know that $\cos\alpha=\cos(\frac{\pi}{2}-\beta)$. According to the co - function identity $\cos(A - B)=\cos A\cos B+\sin A\sin B$, when $A = \frac{\pi}{2}$ and $B=\beta$, $\cos(\frac{\pi}{2}-\beta)=\sin\beta$.

Answer:

D. $\cos\alpha=\sin\beta$