Question

Evaluate $\cot \theta +\frac{1}{\cot \theta }=2$ if $\frac{1}{{\cot }^2\theta }+{\cot }^2\theta =2$

Answer 1

We have,

${\cot }^2\theta +\frac{1}{{\cot }^2\theta }=2$

Adding $2$ both side and we get

${\cot }^2\theta +\frac{1}{{\cot }^2\theta }+2=2+2$

Multiplying and divided by $\cot \theta$

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

$\Rightarrow {(\cot \theta +\frac{1}{\cot \theta })}^2=4$

$\Rightarrow (\cot \theta +\frac{1}{\cot \theta })=\sqrt{4}$

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

Hence proved.