Question

which expression is equivalent to $\csc ^{2}\theta\cos ^{2}\theta$?
a) $\sin ^{2}\theta$
b) $1$
c) $\cos ^{4}\theta$
d) $\cot ^{2}\theta$

Explanation:

Step1: Substitute the reciprocal identity

Recall that \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc^{2}\theta\cos^{2}\theta=\frac{\cos^{2}\theta}{\sin^{2}\theta}\)

Step2: Use the cotangent identity

Since \(\cot\theta = \frac{\cos\theta}{\sin\theta}\), then \(\frac{\cos^{2}\theta}{\sin^{2}\theta}=\cot^{2}\theta\)

Answer:

D. \(\cot^{2}\theta\)