Question

If $xϵ[0,\pi ]$ and $3^{sin2x+2co{s}^{2}x}+3^{1-sin2x+2si{n}^{2}x}=28$, then find number of values of $x$ satisfy given equation.

Answer 1

$3^{\sin 2x+2{\sin }^{2}x}+3^{1-\sin 2x+2{\sin }^{2}x}=28\Rightarrow 3^{\sin 2x+2{\cos }^{2}x}{+3}^{3-(\sin 2x+2{\cos }^{2}x)}=28$
Let $y=3^{\sin 2x+2{\cos }^{2}x}$
$\therefore y=\frac{27}{y}=28$
$\Rightarrow y^2-28y+27=0\Rightarrow y=27$ or $1$
If $y=27$ then $3^{\sin 2x+2{\cos }^{2}x}=3^3$
$\Rightarrow \sin 2x+(2{\cos }^{2}x-1)=0$
$\Rightarrow \sin 2x+\cos 2x=2$
Dividing both sides by $\sqrt{2}$
$\sin (2x+\frac{\pi }{4})=\sqrt{2}$
which is not possible
Now if $y=1$
$\Rightarrow 3^{\sin 2x+2{\cos }^{2}x}=1=3^0\Rightarrow \sin 2x+2{\cos }^{2}x=0\Rightarrow 2\cos x(\sin x+\cos x)=0$
$\therefore \cos x=0$ or $\tan x=-1$