Question

If $\tan \theta +\frac{1}{\tan \theta }=2$, show that ${\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2$.

Answer 1

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

Square both sides of equation (1).

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

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

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

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

Hence, ${\tan }^2\theta +\frac{1}{{\tan }^2\theta }=2$.