Question

If $\sec \theta +\mathrm{cosec}\theta =\sqrt{2}\sec \left(90-\theta \right)$ then what is the value of $\cot \theta$?

• A. $\sqrt{2}$
• B. $2$
• C. $\sqrt{2}-1$
• D. $\sqrt{2}+1$

Answer 1

The correct option is C $\sqrt{2}-1$

Given $\text{sec}\theta -\text{cosec}\theta =\sqrt{2}\text{sec}(90-\theta )$

$⟹\text{sec}\theta =(\sqrt{2}+1)\text{cosec}\theta$

$⟹\tan \theta =\sqrt{2}+1$

$⟹\cot \theta =\frac{1}{\sqrt{2}+1}=\frac{\sqrt{2}-1}{(\sqrt{2}+1)(\sqrt{2}-1)}=\frac{\sqrt{2}-1}{2-1}=\sqrt{2}-1$