<s>[INST] Prove $1 + \cot^2\theta = \csc^2\theta$ Prove the following identity:
$$1 + \cot^2\theta = \csc^2\theta$$
This question is asked because I am curious to know the different ways of proving this identity depending on different characterizations of cotangent and cosecant.
 [/INST] Assuming the First Pythagorean Trigonometric Identity,
$$\sin^2\theta + \cos^2\theta = 1$$
Dividing by $\sin^2\theta$,
$$\Rightarrow \frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}$$ 
$$\Rightarrow \left(\frac{\sin\theta}{\sin\theta}\right)^2 + \left(\frac{\cos\theta}{\sin\theta}\right)^2 = \left(\frac{1}{\sin\theta}\right)^2$$ 
Since $\cot\theta = \large \frac{1}{\tan\theta} = \large\frac{\cos\theta}{\sin\theta}$ and $\csc\theta = \large\frac{1}{\sin\theta}$,
$$\Rightarrow 1 + \cot^2\theta = \csc^2\theta .$$
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