Question

Find Value of x:

$cosecx\times cosec2x=cosec4x$

Answer 1

As $cosecx=\frac{1}{sinx}\Rightarrow sinx\times sin2x=sin4x$

$\Rightarrow sinx\times sin2x=2sin2xcos2x$

$\Rightarrow sin2x(sinx-2cos2x)=0$

$\Rightarrow sin2x(sinx-2(1-2si{n}^{2}x\left)\right)=0$

$\Rightarrow sin2x(4si{n}^{2}x+sinx-2)=0$

$\Rightarrow sin2x=0$ or $sinx=\frac{-1\pm \sqrt{1\pm 32}}{8}$

$\Rightarrow 2sinxcosx=0$

$\Rightarrow cosx=0(sinx\ne 0$ As $cosecx\to \infty )$ or $\Rightarrow sinx=\frac{-1\pm \sqrt{33}}{8}$