How do you evaluate tan (Arc cos (sqrt2/2) )?

$\pm 1$. It is 1, if $a = a r c \cos \left(\frac{\sqrt{2}}{2}\right)$ is restricted to be in the 1st quadrant...
Let $a = a r c \cos \left(\frac{\sqrt{2}}{2}\right) = a r c \cos \left(\frac{1}{\sqrt{2}}\right)$. As cos (+-a)= cos a, a is in the 1st quadrant or in the 4th. Accordingly, $\sin a = \pm \frac{1}{\sqrt{2}} \mathmr{and} , \therefore , \tan a = \pm 1$.
if $a = a r c \cos \left(\frac{\sqrt{2}}{2}\right)$ is restricted to be in the 1st quadrant. the given expression tan a = 1...