# What does -2cos(arctan(6))+csc(arcsec(2)) equal?

Dec 18, 2015

$- 2 \cos \left(\arctan \left(6\right)\right) + \csc \left(\text{arcsec} \left(2\right)\right)$

$= - \frac{2}{\sqrt{37}} + \frac{2}{\sqrt{3}}$

$= \frac{- 6 \sqrt{37} + 74 \sqrt{3}}{111}$

#### Explanation:

$\arctan \left(6\right)$ is one of the angles in a right angled triangle with legs of length $1$, $6$ and hypotenuse $\sqrt{{1}^{2} + {6}^{2}} = \sqrt{37}$

Hence $\cos \left(\arctan \left(6\right)\right) = \frac{1}{\sqrt{37}}$

$\text{arcsec} \left(2\right)$ is one of the angles in a right angled triangle with legs of length $1$, $\sqrt{{2}^{2} - {1}^{2}} = \sqrt{3}$ and hypotenuse $2$.

Hence $\csc \left(\text{arcsec} \left(2\right)\right) = \frac{2}{\sqrt{3}}$

So:

$- 2 \cos \left(\arctan \left(6\right)\right) + \csc \left(\text{arcsec} \left(2\right)\right)$

$= - \frac{2}{\sqrt{37}} + \frac{2}{\sqrt{3}}$

$= - \frac{2 \sqrt{37}}{37} + \frac{2 \sqrt{3}}{3}$

$= \frac{- 6 \sqrt{37} + 74 \sqrt{3}}{111}$