Question

what is the value of cos{cos^-1(-3/2)+π/6}​

Answer 1

Not defined.

$cos^{-1}x$ has domain of definition [-1, 1], so $cos^{-1}\frac{-3}{2}$ isn't defined, and hence the whole thing isn't.

If you meant:

$cos^{-1}\frac{-\sqrt{3}}{2}$, the answer is:

cos(5π/6 + π/4) = -cos15° = -(√3 + 1)/2√2