Question

$if\frac{1}{{\cot }^2\theta }+{\cot }^2\theta =2$ then prove that $\cot \theta +\frac{1}{\cot \theta }=2$

Answer 1

Given that,

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

Adding $2$ both side and we get,

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

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

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

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

Comparing both side and we get,

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

Hence proved.