Question

If ${\tan }^3\theta +{\cot }^3\theta =52$, then the value of ${\tan }^2\theta +{\cot }^2\theta$ is equal to

• A. $14$
• B. $15$
• C. $16$
• D. $17$

Answer 1

The correct option is A $14$

${\tan }^3\theta +{\cot }^3\theta =52$
$(\tan \theta +\cot \theta )({\tan }^2\theta +{\cot }^2\theta -1)=52$
$(\tan \theta +\cot \theta )\left(\right(\tan \theta +\cot \theta {)}^2-3)=52$
Let $\tan \theta +\cot \theta =x$
$x(x^2-3)=52$
$x^3-3x-52=0$
Solving this we get $x=4\Rightarrow \tan \theta +\cot \theta =4$
${\tan }^2\theta +{\cot }^2\theta +2=16=>{\tan }^2\theta +{\cot }^2\theta =14$