Question

Number of solution(s) of the equation $4.{16}^{{\sin }^{2}x}=2^{6\sin x,},$ in $[0,2\pi ],$is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$4.{16}^{{\sin }^{2}x}=2^{6sinx}\Rightarrow 2^2{.2}^{4{\sin }^{2}x}=2^{6\sin x}\Rightarrow 2^{2+4{\sin }^{2}x}=2^{6\sin x}\Rightarrow 2+4{\sin }^{2}x=6\sin x\Rightarrow 4{\sin }^{2}x-6\sin x+2=0\Rightarrow (4\sin x-2)(\sin x-1)=0\Rightarrow \sin x=\frac{1}{2},1$

$x=\frac{\pi }{2},\frac{\pi }{6},\frac{5\pi }{6}$ satisfies the obtained Solution

So, number of solution will be $3$

Therefore, Answer is $3$