Question

The number of solutions for the equation $\sin 2x+\cos 4x=2$ is

• A. $0$
• B. $1$
• C. $2$
• D. Infinite

Answer 1

The correct option is C $2$

Given equation is $\sin 2x+\cos 4x=2$ ....... $\left(i\right)$

We know that $\cos \left(2x\right)=1-2{\sin }^{2}x$

$⟹\sin 2x+\cos 2\left(2x\right)=2$

$⟹\sin 2x+1-2{\sin }^{2}2x=2$

and let $\sin 2x=t$

$⟹-2t^2+t+1=0$

Discriminant of above equation is $1^2-4(-2)=9$ real roots exist.

So, $-2t^2+2t-t+1=0$

$(-2t-1)(t-1)=0$

$\therefore t=-1/2,1$

$\therefore \sin 2x=-1/2$ or $1$

$⟹x={105}^0,{45}^0$

Hence, $2$ solutions exist.