The double angle identity for cosine is
[tex]cos 2 \theta = 2 \cos ^2 \theta -1 [/tex]
so the half angle identity is
[tex]\cos \theta = \pm \sqrt{ \frac 1 2 (\cos 2 \theta + 1) }[/tex]
[tex] \cos \frac{7 \pi}{12} = -\sqrt{ \frac 1 2 (\cos \frac{7 \pi}{6} + 1)} [/tex]
We picked the negative sign because our angle is in the second quadrant, negative cosine.
[tex] \cos \frac{7 \pi}{12} = -\sqrt{ \frac 1 2 (-\cos \frac{ \pi}{6} + 1)} [/tex]
I used the identity [tex]\cos(x+\pi)=-\cos x[/tex]
[tex] \cos \frac{7 \pi}{12} = -\sqrt{ \frac 1 2 (1-\frac{\sqrt 3}{2})}[/tex]
I substituted in there cosine of 30 degrees
[tex] \cos \frac{7 \pi}{12} = -\sqrt{ \dfrac{2 - \sqrt 3}{4}} [/tex]
[tex] \cos \frac{7 \pi}{12} = -\frac 1 2 \sqrt{ 2 - \sqrt 3} [/tex]
Looks like choice d
