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.
[Math] Prove $1 + \cot^2\theta = \csc^2\theta$
alternative-prooftrigonometry
Best Answer
Here we repeat an idea used in the question Prove $\sin^2\theta+\cos^2\theta=1$ but it's slightly different since the functions $\cot$ and $\csc$ aren't defined on $\mathbb R$.
Let $$f(\theta)=\csc^2\theta-\cot^2\theta$$ then $f$ is defined on $\mathbb R\setminus\{k\pi,\; k\in\mathbb Z\}$ and we verify that $f'(\theta)=0$ so $f$ is constant in every interval $(k\pi,(k+1)\pi)$ and we conclude the result from the equality $$f\left(k\pi+\frac{\pi}{2}\right)=1$$