Solve $\displaystyle\csc^2(x)=\csc^2\left(\frac x 2\right)+\csc^2\left(\frac x 4\right)$ , $\forall x\in[0,\pi/2] $.
My attempt: $$\frac{1}{\sin^2\theta}=\frac{1}{\sin^2\frac \theta 2}+\frac{1}{\sin^2\frac \theta 4}$$ then $$1=\frac{\sin^2\theta}{\sin^2\frac \theta 2}+\frac{\sin^2\theta}{\sin^2\frac \theta 4}$$ we have $\theta=2\times\frac \theta 2$ and $\frac \theta 2=2\times\frac \theta4$ then $$1= 4\cos^2 \frac \theta 2+16\cos^2 \frac \theta 2 \cos^2 \frac \theta 4 $$ $$1= \left(4+16\cos^2 \frac \theta 4\right)\cos^2 \frac \theta 2.$$ I'm stuck here. Any help