Find an exact value. cosine of negative seven pi divided by twelve

We want to find cos(-7π/12).

Let x = -(7π)/12.
Then 2x = -(7π)/6.

Note that
cos(2x) = 2cos²x - 1, or
cos x = +/- √([1 + cos(2x)]/2)

Because -(7π)/6 is in the 3rd quadrant, with a reference angle of π/6.
cos(-7π/6) = -cos(π/6) = -√3/2 = -0.866.
Also, -(7π)/12 is in the 2nd quadrant, therefore it's cosine is negative.

Therefore, obtain
cos(-7π/12) = -√[0.5*(1 - 0.866)] = -0.2588.

-0.2588