Question

establish the identity. ((1 + cot^{2}\theta)sin^{2}\theta = 1) rewrite the left side expression by distributing. (sin^{2}\theta+cot^{2}\thetasin^{2}\theta) (do not simplify.) rewrite the second term in the expression from the previous step in terms of sines and cosines. (sin^{2}\theta+squaresin^{2}\theta)

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 $\sin^{2}\theta+\frac{\cos^{2}\theta}{\sin^{2}\theta}\sin^{2}\theta$.

Step3: Simplify the expression

$\frac{\cos^{2}\theta}{\sin^{2}\theta}\sin^{2}\theta=\cos^{2}\theta$, 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:

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