# How do you solve and find the value of cos(cos^-1(sqrt2/2)-pi/2)?

Apr 7, 2018

$\frac{\sqrt{2}}{2}$

#### Explanation:

First, recall that $\cos \left(x - \frac{\pi}{2}\right) = \sin x$, so here, we're truly being asked to find

$\sin \left({\cos}^{-} 1 \left(\frac{\sqrt{2}}{2}\right)\right)$

Now, determine ${\cos}^{-} 1 \left(\frac{\sqrt{2}}{2}\right)$.

$x = {\cos}^{-} 1 \left(\frac{\sqrt{2}}{2}\right) \Leftrightarrow \cos x = \frac{\sqrt{2}}{2}$, from the definition of an inverse function.

Keeping in mind that the domain of the inverse cosine is $\left[- 1 , 1\right] ,$ the only solution to the above equation is

$x = \frac{\pi}{4}$

Thus, we get

$\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$