Answer
$$\frac{\sin^2x}{\cos^2x}+\sin x\csc x=\sec^2x$$
Work Step by Step
$$A=\frac{\sin^2x}{\cos^2x}+\sin x\csc x$$
$$A=\Big(\frac{\sin x}{\cos x}\Big)^2+\sin x\csc x$$
- Quotient Identity: $$\frac{\sin x}{\cos x}=\tan x$$
- Reciprocal Identity:
$$\csc x=\frac{1}{\sin x}$$
Replace them into $A$, we have
$$A=\tan^2 x+\sin x\times\frac{1}{\sin x}$$
$$A=\tan^2 x+1$$
- Pythagorean Identity:
$$\tan^2 x+1=\sec^2 x$$
which means $$A=\sec^2x$$
