<s>[INST] How to have idea to prove trigonometric identities Hi to explain this better I'll take an example.
I have this identity that's giving me a hard time.
$$\frac{\cos^2(a)-\sin^2(b)}{\sin^2(a)\sin^2(b)} = \cot^2(a)\cot^2(b)-1$$
This is what i would do
$$\cos^2(a)/(\sin^2(a)·\sin^2(b))-\sin^2(b)/(\sin^2(a)·\sin^2(b)) \\ 
\cot^2(a)·1/\sin^2(b)-1/\sin^2(a)$$
then, we know that
$$1=\cos^2(b)+\sin^2(b) \\
\vdots \\
\cot^2(a)·\cot^2(b)+\cot^2(a)-1/\sin^2(a)$$
Which of course is wrong but just wanted to show you guys how my mind thinks.
Is there any right way of solving this or do I just  have to keep trying.
THANKS.
 [/INST] First use $\cot x=\frac{\cos x}{\sin x}$ and work with the RHS .
$$\cot^2 a \cdot \cot^2 b-1=\frac{\cos^2 a \cdot \cos^2 b}{\sin^2 a \cdot \sin^2 b}-1=\frac{\cos^2a \cdot \cos^2 b-\sin^2 a \cdot \sin^2 b}{\sin^2 a \cdot \sin^2 b}$$ 
All that remains to prove is that 
$$\cos^2a \cdot \cos^2 b-\sin^2 a \cdot \sin^2 b=\cos^2 a-\sin^2 b$$ but this follows using the classical $\cos^2x+\sin^2x=1$ as follows :
$$\cos^2a \cdot \cos^2 b-\sin^2 a \cdot \sin^2 b=\cos^2 a \cdot (1-\sin^2 b)-(1-\cos^2a) \cdot \sin^2 b=\cos^2 a-\sin^2 b$$ as wanted .
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