Prove that: $$\mathrm{cosec}\frac {\alpha}{8}+\mathrm{cosec}\frac {\alpha}{4}+\mathrm{cosec}\frac {\alpha}{2}=\cot \frac {\alpha}{16} - \cot \frac {\alpha}{2}$$.
My Attempt:
$$\text{L.H.S}=\mathrm{cosec}\frac {\alpha}{8}+\mathrm{cosec}\frac {\alpha}{4} + \mathrm{cosec}\frac {\alpha}{2}$$ $$=\frac {1}{\sin(\alpha/8)} + \frac {1}{2\sin(\alpha/8)\cdot \cos(\alpha/8)} + \frac {1}{\sin (\alpha/2)}$$. $$=\frac {2\cos (\alpha/8) +1}{2\sin(\alpha/8)\cdot\cos(\alpha/8)} + \frac {1}{\sin(\alpha/2)}$$.
How should I move on further? Please help.