Question

establish the identity.
(1 + cot²θ)sin²θ = 1

rewrite the left side expression by distributing.
sin²θ + cot²θsin²θ (do not simplify.)
rewrite the second term in the expression from the previous step in terms of sines and cosines.
sin²θ + ( )sin²θ

Explanation:

Step1: Recall cotangent identity

We know that $\cot\theta=\frac{\cos\theta}{\sin\theta}$, so $\cot^{2}\theta = \frac{\cos^{2}\theta}{\sin^{2}\theta}$.

Step2: Substitute cotangent identity

Substitute $\cot^{2}\theta=\frac{\cos^{2}\theta}{\sin^{2}\theta}$ into $\sin^{2}\theta+\cot^{2}\theta\sin^{2}\theta$. We get $\sin^{2}\theta+\frac{\cos^{2}\theta}{\sin^{2}\theta}\times\sin^{2}\theta$.

Step3: Simplify the expression

$\frac{\cos^{2}\theta}{\sin^{2}\theta}\times\sin^{2}\theta=\cos^{2}\theta$, so the expression becomes $\sin^{2}\theta+\cos^{2}\theta$.

Step4: Use Pythagorean identity

By the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$.

Answer:

1