Question

Solve the following equations.
$sin\frac{x}{2}cos2x+si{n}^{2}xcos\frac{x}{2}=co{s}^{2}xcos\frac{x}{2}.$

Answer 1

$sin\frac{x}{2}.cos2x+si{n}^{2}x.cos\frac{x}{2}=co{s}^{2}xcos\frac{x}{2}$

$\Rightarrow sin\frac{x}{2}cos2x=co{s}^{2}xcos\frac{x}{2}-si{n}^{2}x.cos\frac{x}{2}$

$\Rightarrow sin\frac{x}{2}cos2x=cos\frac{x}{2}(co{s}^{3}x-si{n}^{2}x)$

$\Rightarrow sin\frac{x}{2}.co{s}^{2}x=cos\frac{x}{2}.cos2x$

$\Rightarrow cos2x[sin\frac{x}{2}-cos\frac{x}{2}]=0$

$cos2x=0$ or $sin\frac{x}{2}-cos\frac{x}{2}=0$

$2x=(2n+1)\pi /2$ or $tan\frac{x}{2}=1$

$x=(2n+1)\pi /4$

$\frac{x}{2}=x\pi +\pi /4$

$x=2n\pi +\pi /2$