If tan x
Question:

If $\tan x=-\frac{1}{\sqrt{5}}$ and $\theta$ lies in the IV quadrant, then the value of $\cos x$ is

(a) $\frac{\sqrt{5}}{\sqrt{6}}$

(b) $\frac{2}{\sqrt{6}}$

(c) $\frac{1}{2}$

(d) $\frac{1}{\sqrt{6}}$

Solution:

(a) $\frac{\sqrt{5}}{\sqrt{6}}$

In the fourth quadrant, $\cos x$ and $\sec x$ are positive.

$\cos x=\frac{1}{\sec x}$

$=\frac{1}{\sqrt{\sec ^{2} x}}$

$=\frac{1}{\sqrt{1+\tan ^{2} x}}$

$=\frac{1}{\sqrt{1+\left(-\frac{1}{\sqrt{5}}\right)^{2}}}$

$=\frac{1}{\sqrt{\frac{6}{5}}}$

$=\frac{\sqrt{5}}{\sqrt{6}}$