Question

If $\tan \theta +\cot \theta =5$, then ${\tan }^2\theta +{\cot }^2\theta =$

• A. $27$
• B. $29$
• C. $24$
• D. $23$

Answer 1

The correct option is D $23$

$\tan \theta +\cot \theta =5$

$\Rightarrow (\tan \theta +\cot \theta {)}^2=5^2$, square both sides
$\Rightarrow {\tan }^2\theta +2\tan \theta \cdot \cot \theta +{\cot }^2\theta =25$
$\Rightarrow {\tan }^2\theta +{\cot }^2\theta +2=25(\because \tan \theta \cdot \cot \theta =1)$
$\Rightarrow {\tan }^2\theta +{\cot }^2\theta =23$