Question

Double-Angle and Half-Angle Identiies [See Attachment] Question2

Answer 1

The double angle identity for cosine is

`cos 2 \theta = 2 \cos ^2 \theta -1`

so the half angle identity is

`\cos \theta = \pm √( \frac 1 2 (\cos 2 \theta + 1) )`

`\cos (7 \pi)/(12) = -\sqrt{ \frac 1 2 (\cos (7 \pi)/(6) + 1)}`

We picked the negative sign because our angle is in the second quadrant, negative cosine.

`\cos (7 \pi)/(12) = -\sqrt{ \frac 1 2 (-\cos ( \pi)/(6) + 1)}`

I used the identity
`\cos(x+\pi)=-\cos x`

`\cos (7 \pi)/(12) = -\sqrt{ \frac 1 2 (1-(\sqrt 3)/(2))}`

I substituted in there cosine of 30 degrees

`\cos (7 \pi)/(12) = -\sqrt{ (2 - \sqrt 3)/(4)}`

`\cos (7 \pi)/(12) = -\frac 1 2 √( 2 - \sqrt 3)`

Looks like choice d