Question

Formula of $cot\left(3x\right)$ is

Answer 1

Estimate the formula:

The given expression is,

$cot\left(3x\right)=cot\left(2x+x\right)$

$=\frac{cot2xcotx-1}{cot2x+cotx}$ $\left[\because cot\left(x+y\right)=\frac{cotxcoty-1}{cotx+coty}\right]$

$=\frac{\left({\displaystyle \frac{co{t}^{2}x-1}{2cotx}}\right)cotx-1}{\left({\displaystyle \frac{co{t}^{2}x-1}{2cotx}}\right)+cotx}$ $\left[\because cot\left(2x\right)=\frac{co{t}^{2}x-1}{2cotx}\right]$

$=\frac{{\displaystyle \frac{co{t}^{3}x-cotx-2cotx}{2cotx}}}{{\displaystyle \frac{co{t}^{2}x-1+2co{t}^{2}x}{2cotx}}}$

$=\frac{co{t}^{3}x-3cotx}{3co{t}^{2}x-1}$

$=\frac{3cotx-co{t}^{3}x}{1-3co{t}^{2}x}$

Hence, the formula of $cot\left(3x\right)$ is $\frac{3cotx-co{t}^{3}x}{1-3co{t}^{2}x}$.