Question

establish the identity. (1 + cot²θ) sin²θ = 1
rewrite the left side expression by distributing.
sin²θ + cot²θ sin²θ (do not simplify.)
rewrite the second term in the expression from the previous step in terms of sines and cosines.
sin²θ + (cos²θ / sin²θ) sin²θ
simplify the expression from the previous step by canceling the common factors.
sin²θ +

Explanation:

Step1: Distribute the left - hand side

$(1 + \cot^{2}\theta)\sin^{2}\theta=\sin^{2}\theta+\cot^{2}\theta\sin^{2}\theta$

Step2: Rewrite cotangent in terms of sine and cosine

Since $\cot\theta=\frac{\cos\theta}{\sin\theta}$, then $\cot^{2}\theta=\frac{\cos^{2}\theta}{\sin^{2}\theta}$, and $\sin^{2}\theta+\cot^{2}\theta\sin^{2}\theta=\sin^{2}\theta+\frac{\cos^{2}\theta}{\sin^{2}\theta}\sin^{2}\theta$

Step3: Simplify the second term

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

Step4: Use the Pythagorean identity

We know that $\sin^{2}\theta+\cos^{2}\theta = 1$

Answer:

1