Find the x-coordinate of the stationary point on the curve $\tan(x)\cos(2x)$ for $0 < x < \pi/2$

Question

Can someone please show me how to find the x-coordinate for the stationary point for this curve?

$y=\tan(x)\cos(2x)$ for $0 < x < \pi/2$

This is what I've done so far:

$$\frac{dy}{dx}=\cos(2x)\sec^2(x)-2\tan(x)\sin(2x)$$

$$\cos(2x)\sec^2(x)-2\tan(x)\sin(2x)=0$$ $$1-\tan^2(x)=4\sin^2(x)$$

I don't know how to continue this to find the answer.

The final answer should be $x\approx 0.452$

Answer 1

$\cos (2x)=2\tan x \sin (2x)\cos^{2}x$. So $\cos (2x)=\sin^{2} (2x)=1-\cos^{2} (2x)$. Solve this quadratic for $\cos (2x)$.