Question

If $\tan \theta +\cot \theta =2{\sin }^2\theta +2{\cos }^2\theta$ then find ${\tan }^4\theta +{\cot }^4\theta$.

Answer 1

Given $\tan \theta +\cot \theta =2{\sin }^2\theta +2{\cos }^2\theta$

$⟹\tan \theta +\cot \theta =2$

Squaring on both sides

$(\tan \theta +\cot \theta {)}^2=2^2$

${\tan }^2\theta +{\cot }^2\theta +2\tan \theta \cot \theta =4$

${\tan }^2\theta +{\cot }^2\theta +2=4$

$⟹{\tan }^2\theta +{\cot }^2\theta =2$

Squaring on both sides

$({\tan }^2\theta +{\cot }^2\theta {)}^2=2^2$

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

${\tan }^4\theta +{\cot }^4\theta =2+4$

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