Sample problem 1

$x = sin(\frac{3\pi}{4}) + \frac{1}{2}sec(\frac{13\pi}{3})$
$x = \frac{\sqrt{2}}{2} + \frac{1}{2}\frac{1}{cos(\frac{13\pi}{3})}$
$x = \frac{\sqrt{2}}{2} + \frac{1}{2}\frac{1}{cos(4\pi + \frac{\pi}{3})}$
$x = \frac{\sqrt{2}}{2} + \frac{1}{2}\frac{1}{cos(\frac{\pi}{3})}$
$x = \frac{\sqrt{2}}{2} + \frac{1}{2}\frac{1}{\frac{1}{2}}$
$x = 1 + \frac{\sqrt{2}}{2}$
$x = \frac{2 + \sqrt{2}}{2}$ -- optional

Sample problem 2

$x = cot^2(\frac{41\pi}{6}) - csc^2(\frac{37\pi}{3})$

$x = cot^2\left(6 + \frac{5\pi}{6}\right) - csc^2\left(12 + \frac{\pi}{3}\right)$

$x = cot^2(\frac{5\pi}{6}) - csc^2(\frac{\pi}{3})$

$x = \left(\frac{cos\left(\frac{5\pi}{6}\right)}{sin\left(\frac{5\pi}{6}\right)}\right)^2 - \frac{1}{\left(sin\left(\frac{\pi}{3}\right)\right)^2}

$x = \left(\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right)^2 - \frac{1}{\left(\frac{\sqrt{3}}{2}\right)^2}$

$x = 3 - \frac{1}{\frac{3}{4}}$

$x = 3 - \frac{4}{3}$

$x = \frac{5}{3}$