Question

If $\cot A+\frac{1}{\cot A}=2,$ find the value of ${\cot }^{2}A+\frac{1}{{\cot }^{2}A.}$

Answer 1

Given: $\cot A+\frac{1}{\cot A}=2$

Squaring both sides, we get

$\Rightarrow (\cot A+\frac{1}{\cot A})=2^2$

$\Rightarrow {\cot }^{2}A+(\frac{1}{\cot A}{)}^2+2\cot A\times \frac{1}{\cot A}=4$ [$\because (a+b{)}^2=a^2+b^2+2ab$]

$\Rightarrow {\cot }^{2}A+\frac{1}{{\cot }^{2}A}+2=4$

$\Rightarrow {\cot }^{2}A+\frac{1}{{\cot }^{2}A}=4-2$

$\Rightarrow {\cot }^{2}A+\frac{1}{{\cot }^{2}A}=2$

Hence, the value of ${\cot }^{2}A+\frac{1}{{\cot }^{2}A}$ is $2.$