Answer
$$\cot^2\theta(1+\tan^2\theta)=\csc^2\theta$$
Work Step by Step
$$A=\cot^2\theta(1+\tan^2\theta)$$
- Pythagorean Identity:
$$\tan^2\theta+1=\sec^2\theta$$
Replace into $A$:
$$A=\cot^2\theta\times\sec^2\theta$$
- Quotient Identity:
$$\cot\theta=\frac{\cos\theta}{\sin\theta}$$
- Reciprocal Identity:
$$\sec\theta=\frac{1}{\cos\theta}$$
Replace them into $A$:
$$A=\frac{\cos^2\theta}{\sin^2\theta}\times\frac{1}{\cos^2\theta}$$
$$A=\frac{1}{\sin^2\theta}=\Big(\frac{1}{\sin\theta}\Big)^2$$
$$A=\csc^2\theta\hspace{1cm}\text{(Reciprocal Identity)}$$
