# Chapter 5 - Quiz (Sections 5.1-5.4) - Page 230: 2

$$\cot^2x+\csc^2x=\frac{\cos^2x+1}{\sin^2x}$$

#### Work Step by Step

$$A=\cot^2 x+\csc^2 x$$ From Pythagorean Identity, we can rewrite $\csc^2 x$ as follows: $$\csc^2 x=1+\cot^2 x$$ So $$A=\cot^2x+1+\cot^2x$$ $$A=2\cot^2x+1$$ Now we rewrite $\cot x$ using Quotient Identity: $$\cot x=\frac{\cos x}{\sin x}$$ Therefore, $$A=2\Big(\frac{\cos x}{\sin x}\Big)^2+1$$ $$A=\frac{2\cos^2 x}{\sin^2x}+1$$ $$A=\frac{2\cos^2x+\sin^2x}{\sin^2x}$$ $$A=\frac{\cos^2x+(\cos^2 x+\sin^2 x)}{\sin^2x}$$ Finally, recall that $\cos^2x+\sin^2x=1$ $$A=\frac{\cos^2x+1}{\sin^2x}$$ Overall, $$\cot^2x+\csc^2x=\frac{\cos^2x+1}{\sin^2x}$$

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