Question

verify the identity.
\frac{\csc \theta \sin \theta}{\cot \theta}=\tan \theta

to verify the identity, start with the more complicated side and transform it to look like the other side. choose the correct transformations and transform
\frac{\csc \theta \sin \theta}{\cot \theta}=\frac{\csc \theta \square}{\cot \theta}
=\frac{\square}{\cot \theta}
=\tan \theta

Explanation:

Step1: Substitute \(\sin\theta\)

We know that \(\csc\theta=\frac{1}{\sin\theta}\), so \(\frac{\csc\theta\sin\theta}{\cot\theta}=\frac{\csc\theta\times\sin\theta}{\cot\theta}\). Since \(\csc\theta\times\sin\theta = 1\), the expression becomes \(\frac{1}{\cot\theta}\).

Step2: Use the reciprocal identity

We know that \(\tan\theta=\frac{1}{\cot\theta}\).

Answer:

\(\frac{\csc\theta\sin\theta}{\cot\theta}=\frac{\csc\theta\times\sin\theta}{\cot\theta}=\frac{1}{\cot\theta}=\tan\theta\)