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