Question

If $5\cos \theta \sin \theta =1,$ then the value of ${\tan }^3\theta +{\cot }^3\theta$ is

• A. $110$
• B. $115$
• C. $125$
• D. $100$

Answer 1

The correct option is A $110$

Given $5\cos \theta \sin \theta =1$
$\Rightarrow 5\cos \theta \sin \theta ={\sin }^2\theta +{\cos }^2\theta \Rightarrow 5=\tan \theta +\cot \theta \Rightarrow (\tan \theta +\cot \theta {)}^3=125\Rightarrow {\tan }^3\theta +{\cot }^3\theta +3\tan \theta \cot \theta (\tan \theta +\cot \theta )=125\Rightarrow {\tan }^3\theta +{\cot }^3\theta +3\times 5=125\therefore {\tan }^3\theta +{\cot }^3\theta =110$