Question

Решите уравнение $cos x = (cosfrac{x}{2}$ - sin $frac{x}{2}$) ^2 - 1

Answer 1

$cos(x)=[cos( frac{x}{2} )-sin( frac{x}{2} )]^2-1$
$cos(x)=[sin( frac{pi}{2}+ frac{x}{2} )-sin( frac{x}{2} )]^2-1$
$cos(x)=[2sin( frac{ frac{pi}{2} + frac{x}{2}- frac{x}{2}}{2} )cos(frac{ frac{pi}{2} + frac{x}{2}+ frac{x}{2}}{2})]^2-1;$
$cos(x)=[2cos(frac{pi}{4} + frac{x}{2})sin( frac{pi}{4} )]^2-1;$
$cos(x)+1=2cos^2(frac{pi}{4} + frac{x}{2});$
$cos(x)+1=2*frac{1+cos(frac{pi}{2} + x)}{2}$
$cos(x)+1=1+cos(frac{pi}{2} + x)$
$cos(x)+cos(frac{pi}{2} + x)=0;$
$2cos( frac{frac{pi}{2}+ x-x}{2} )*cos( frac{frac{pi}{2}+ x+x}{2} )=0$
$cos( frac{pi}{4}+ x )=0$
$frac{pi}{4}+ x = frac{pi}{2}+pi n,nin Z$
$x = frac{pi}{4}+pi n,nin Z$

Ответ: $frac{pi}{4}+pi n,nin Z$