Question

If cosec x + cot x = α, then sin x = _________.

Answer 1

Given cosec x + cot x = a (1)
Since cosec²x – cot²x = 1
(cosec x – cot x) (cosec x + cot x) = 1
i.e (cosec x – cot x) a = 1
$\mathrm{i}.\mathrm{e}\mathrm{cosec}x-\cot x=\frac{1}{a}\left(2\right)$
adding (1) and (2)
$$\begin{gathered} 2\mathrm{cosec}x=a+\frac{1}{a} \\ 2\mathrm{cosec}x=\frac{a^2+1}{a} \\ \frac{2a}{a^2+1}=\frac{1}{\mathrm{cosec}x} \\ \mathrm{i}.\mathrm{e}\sin x=\frac{2a}{a^2+1} \end{gathered}$$