What does cos(arctan(2))-sin(arcsec(5)) equal?

May 19, 2016

$\cos \left(\arctan 2\right) - \sin \left(a r c \sec 5\right)$ is $\pm 1.427$ or $\pm 0.5326$

Explanation:

$\cos \left(\arctan 2\right) - \sin \left(a r c \sec 5\right)$

= $\cos \alpha - \sin \beta$, where

$\tan \alpha = 2$ and $\sec \beta = 5$

Now, $\cos \alpha = \frac{i}{\sec} \alpha = \frac{1}{\sqrt{1 + {\tan}^{2} \alpha}}$

= $\frac{1}{\sqrt{1 + {2}^{2}}} = \pm \frac{1}{\sqrt{5}}$

and $\sin \beta = \sqrt{1 - {\cos}^{2} \beta} = \sqrt{1 - \frac{1}{\sec} ^ 2 \beta}$

= $\sqrt{1 - \frac{1}{5} ^ 2} = \sqrt{1 - \frac{1}{25}} = \sqrt{\frac{24}{25}} = \pm \frac{2}{5} \sqrt{6}$

Hence $\cos \alpha - \sin \beta = \left(\pm \frac{1}{\sqrt{5}}\right) - \left(\pm \frac{2}{5} \sqrt{6}\right)$

= $\left(\pm 0.4472\right) - \left(\pm 0.9798\right)$

Taking different sign combinations can take four different values of $\cos \alpha - \sin \beta$, which are $\pm 1.427$ or $\pm 0.5326$