Question

If $\tan \alpha +\cot \alpha =p$ where $p\ge 2$, then the value of $\tan \alpha -\cot \alpha$ is

• A. $p^3-4$
• B. $4\sqrt{p^2-4}$
• C. $\sqrt{p^2-2}$
• D. $\sqrt{p^2-4}$

Answer 1

The correct option is D $\sqrt{p^2-4}$

Let $\tan \alpha -\cot \alpha =x$

Squaring, ${\tan }^2\alpha +{\cot }^2\alpha -2\tan \alpha \cot \alpha =x^2$ .... (1)

Given $\tan \alpha +\cot \alpha =p$

Squaring, ${\tan }^2\alpha +{\cot }^2\alpha +2\tan \alpha \cot \alpha =p^2$

${\tan }^2\alpha +{\cot }^2\alpha =p^2-2$ ...$(\tan \alpha \cot \alpha =1)$

Put in (1), we get
$p^2-2-2=x^2\Rightarrow x=\sqrt{p^2-4}$