Given is [tex] 2*cos^{2} (x)=sin(2x) [/tex]
We can use following formulas to solve this problem :-
1. [tex] \frac{sin\;\theta}{cos\;\theta} = tan\;\theta [/tex]
2. sin(2∅) = 2·sin(∅)·cos(∅)
Solving the given equation :-
[tex] 2*cos^{2} (x)=sin(2x) \\\\2*cos^{2} (x)=2*sin(x)*cos(x) \\\\2*cos^{2} (x)-2*sin(x)*cos(x)=0 \\\\2*cos^{2}(x)*(1-\frac{sin(x)}{cos(x)}) =0 \\\\2*cos^{2}(x)*(1-tan(x)) =0 \\\\cos^{2}(x) = 0 \;or\; (1-tan(x)) =0 \\\\cos(x) = 0 \;or\; tan(x) =1 \\\\x = cos^{-1}(0) \;or\; x=tan^{-1}(1) \\\\x=\frac{\pi}{2} \;or\; x=\frac{\pi}{4} [/tex]
Hence, final answer is [tex] x=\frac{\pi}{2} \;or\; x=\frac{\pi}{4} [/tex].
