Question

If $\sec \theta +\tan \theta =k$, then $\cos \theta =$ _______.

• A. $\frac{k^2+1}{2k}$
• B. $\frac{2k}{k^2+1}$
• C. $\frac{k}{k^2+1}$
• D. $\frac{k}{k^2-1}$

Answer 1

The correct option is B $\frac{2k}{k^2+1}$

Given $\text{sec}\theta +\tan \theta =k$

As we know that

${\text{sec}}^2\theta -{\tan }^2\theta =1$

$⟹(\text{sec}\theta +\tan \theta )(\text{sec}\theta -\tan \theta )=1$

$⟹\text{sec}\theta -\tan \theta =\frac{1}{k}$

$⟹\text{sec}\theta =\frac{k+\frac{1}{k}}{2}=\frac{1+k^2}{2k}$

$⟹\cos \theta =\frac{2k}{k^2+1}$