Question

If $\tan \theta +\frac{1}{\tan \theta }=2$, prove ${\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2$

Answer 1

Given $\tan \theta +\frac{1}{\tan \theta }=2$

squaring on both sides

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

$⟹{\tan }^2\theta +\frac{1}{{\tan }^2\theta }+2\left(\tan \theta \right)(\frac{1}{\tan \theta })=4$

$⟹{\tan }^2\theta +\frac{1}{{\tan }^2\theta }+2=4$

$⟹{\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2$