Question

Precalculus PLEASE HELP ME right answers only check all please

Answer 1

Given :

$cos(x)cos(\frac{\pi }{7}) + sin(x)sin(\frac{\pi }{7})=-\frac{\sqrt{2}}{2}$

We know the identity

$cos(x)cos(y) + sin(x)sin(y)=cos(x-y)$

We use this property to simplify the left hand side

So  $cos(x)cos(\frac{\pi }{7}) + sin(x)sin(\frac{\pi }{7})=cos(x - \frac{\pi }{7})$

$cos(x - \frac{\pi }{7}) =-\frac{\sqrt{2}}{2})$

we know ,

when $cos(x) =-\frac{\sqrt{2}}{2})$ then

$x = \frac{3\pi }{4} , \frac{5\pi }{4}$

For  $cos(x - \frac{\pi }{7}) =-\frac{\sqrt{2}}{2})$

$x - \frac{\pi }{7}= \frac{3\pi }{4} and x - \frac{\pi }{7}= \frac{5\pi }{4}$

Add $\frac{\pi }{7}$ on both sides

$x= \frac{3\pi }{4} + \frac{\pi }{7} and x= \frac{5\pi }{4} +\frac{\pi }{7}$

Finally we add 2npi for general solution

So options C  and D  are correct