Question

Solve the following equations.
$co{s}^{2}x-cosx=4sinx-sin2x.$

Answer 1

$co{s}^{2}x-cosx=4sinx-sin2x$

$1-si{n}^{2}x-4sinx=cosx-2sinx.cos$

$co{s}^{2}x-2cosx=4sinx-2sin2x$

$\Rightarrow co{s}^{2}x-2cosx=4sinx-2sin.cosx$

$\Rightarrow cosx(cosx-2)=2sinx(2-cosx)$

$\Rightarrow (cosx-2)(cosx+2sinx)=0$

$\therefore cosx=2$ or $cosx=-2sinx$

$\Rightarrow$ not possible for any value of s (No solution)

$\Rightarrow cost=x=-2$

$\Rightarrow tanx=-2^{-1}$

$\Rightarrow x=ta{n}^{-1}(-2{)}^{+2}+n\pi$

$\therefore x=n\pi +ta{n}^{-1}(-2{)}^{-1}=n\pi +ta{n}^{-1}(1/2);nϵz$