Given :
[tex]cos(x)cos(\frac{\pi }{7}) + sin(x)sin(\frac{\pi }{7})=-\frac{\sqrt{2}}{2}[/tex]
We know the identity
[tex]cos(x)cos(y) + sin(x)sin(y)=cos(x-y)[/tex]
We use this property to simplify the left hand side
So [tex]cos(x)cos(\frac{\pi }{7}) + sin(x)sin(\frac{\pi }{7})=cos(x - \frac{\pi }{7})[/tex]
[tex]cos(x - \frac{\pi }{7}) =-\frac{\sqrt{2}}{2})[/tex]
we know ,
when [tex]cos(x) =-\frac{\sqrt{2}}{2})[/tex] then
[tex]x = \frac{3\pi }{4} , \frac{5\pi }{4}[/tex]
For [tex]cos(x - \frac{\pi }{7}) =-\frac{\sqrt{2}}{2})[/tex]
[tex]x - \frac{\pi }{7}= \frac{3\pi }{4} and x - \frac{\pi }{7}= \frac{5\pi }{4}[/tex]
Add [tex]\frac{\pi }{7}[/tex] on both sides
[tex]x= \frac{3\pi }{4} + \frac{\pi }{7} and x= \frac{5\pi }{4} +\frac{\pi }{7}[/tex]
Finally we add 2npi for general solution
So options C and D are correct
