## Algebra 2 Common Core

$$\sin\theta$$
Simplify $$\csc\theta-\cos\theta\cot\theta.$$ Use the Reciprocal Identity $$\csc\theta=\frac{1}{\sin\theta}$$ and the Cotangent Identity $$\cot\theta=\frac{\cos\theta}{\sin\theta}$$ to obtain $$\csc\theta-\cos\theta\cot\theta=\bigg(\frac{1}{\sin\theta}\bigg)-\cos\theta\bigg(\frac{\cos\theta}{\sin\theta}\bigg)$$ $$=\frac{1-\cos^{2}\theta}{\sin\theta}.$$ Use the Pythagorean Identity $$\sin^{2}\theta+\cos^{2}\theta=1$$ rearranged as $$1-\cos^{2}\theta=\sin^{2}\theta$$ to produce $$\csc\theta-\cos\theta\cot\theta=\frac{\sin^{2}\theta}{\sin\theta}=\sin\theta.$$