Trigonometry question (given that cosec x = 9)

Question

Given that cosec x = 9

without using a calculator, evaluate
(a) cot x (b) tan x, and (c) cos x

Answer 1

We see $\sin x = 1/9, \cos x = \sqrt{1 - 1/81}= \sqrt{80/81}$ if $x$ is in 1st quadrant and $-\sqrt{80/81}$ if $x$ is in 2nd quadrant. Then $\tan x$ and $\cot x$ can be determined from $\sin x$ and $\cos x$.

Answer 2

$$\operatorname{cosec}(x)=9\implies \sin(x)=\frac19$$

$$\sin^2(x)+\cos^2(x)=1\implies \cos(x)=\pm\frac{\sqrt{80}}{9}$$

$$\tan(x)=\frac{\sin(x)}{\cos(x)} =\pm\frac{1}{\sqrt{80}}$$

$$\cot(x)=\frac{1}{\tan(x)}=\pm \sqrt{80}$$