How to Prove 1+cot^2(theta)=cosec^2(theta)

  

Use the Pythagorean identity:

$$\sin^2\theta + \cos^2\theta = 1$$see how to prove it

Divide both sides of the identity by ${\sin }^2(\theta )$

$$\frac{{\sin }^2(\theta )}{{\sin }^2(\theta )}+\frac{{\cos }^2(\theta )}{{\sin }^2(\theta )}=\frac{1}{{\sin }^2(\theta )<span\; face="Arimo,\; sans-serif"> </span>}$$

Final Answer:

$$\boxed{{\displaystyle {\mathrm{1\; +\; cot}}^2(\theta )\mathrm{<br>}={\mathrm{cosec}}^2(\theta )}}$$

This proves the identity.