Решите уравнение [latex] cos x = (cos\frac{x}{2} [/latex] - sin [latex] \frac{x}{2} [/latex]) ^2 - 1

[latex]cos(x)=[cos( \frac{x}{2} )-sin( \frac{x}{2} )]^2-1[/latex] [latex]cos(x)=[sin( \frac{\pi}{2}+ \frac{x}{2} )-sin( \frac{x}{2} )]^2-1[/latex] [latex]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; [/latex] [latex]cos(x)=[2cos(\frac{\pi}{4} + \frac{x}{2})sin( \frac{\pi}{4} )]^2-1;[/latex] [latex]cos(x)+1=2cos^2(\frac{\pi}{4} + \frac{x}{2});[/latex] [latex]cos(x)+1=2*\frac{1+cos(\frac{\pi}{2} + x)}{2} [/latex] [latex]cos(x)+1=1+cos(\frac{\pi}{2} + x)[/latex] [latex]cos(x)+cos(\frac{\pi}{2} + x)=0;[/latex] [latex]2cos( \frac{\frac{\pi}{2}+ x-x}{2} )*cos( \frac{\frac{\pi}{2}+ x+x}{2} )=0[/latex] [latex]cos( \frac{\pi}{4}+ x )=0[/latex] [latex]\frac{\pi}{4}+ x = \frac{\pi}{2}+\pi n,n\in Z [/latex] [latex]x = \frac{\pi}{4}+\pi n,n\in Z [/latex] Ответ: [latex]\frac{\pi}{4}+\pi n,n\in Z [/latex]