<s>[INST] How to prove in this trig problem i have to prove this
$$\frac{\cos 3x}{\sin 2x \sin 4x}+\frac{\cos 5x}{\sin 4x \sin 6x}+\frac{\cos 7x}{\sin 6x \sin 8x}+\frac{\cos 9x}{\sin 8x \sin 10x} = \frac{1}{2}\csc x(\csc 2x - \csc 10x)$$
i tried taking lcm but does not leads to anything. i believe i have to write numerator as something in terms of denominator which i do not know how. Help. 
Thanks
 [/INST] Hint: Observe
\begin{align}
\frac{\cos 3x}{\sin 2 x \sin 4x } = \frac{1}{2\sin x}\left(\frac{1}{\sin 2x}-\frac{1}{\sin 4x} \right)
\end{align}
since
\begin{align}
\frac{1}{\sin 2x}-\frac{1}{\sin 4x}=&\  \frac{\sin 4x-\sin 2x}{\sin 2x \sin 4x}\\
=&\ \frac{\sin 3x\cos x+\sin x\cos 3x-\sin 3x \cos x+\sin x\cos 3x}{\sin 2x \sin 4x}\\
=&\ \frac{2\cos 3x \sin x}{\sin 2x \sin 4x}.
\end{align}
Telescoping sum. 
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