Question

If $tan \left(\frac{\theta}{2}\right)=\frac{2}{3}$, then $sec\,\theta $ is equal to

• A. $\frac{13 }{ 5}$
• B. $\frac{ 13}{ 3}$
• C. $\frac{3 }{13 }$
• D. $\frac{ 5}{13 }$
• E. $\frac{12 }{13 }$

Answer 1

Answer: (A)

Given, $\tan \left(\frac{\theta}{2}\right)=\frac{2}{3}$
$\because\, \tan\, \theta=\frac{2 \tan \frac{\theta}{2}}{1-\tan ^{2} \frac{\theta}{2}}=\frac{2 \times \frac{2}{3}}{1-\frac{4}{9}}$
$=\frac{4}{3} \times \frac{9}{5}$
$\Rightarrow \, \tan \theta=\frac{12}{5} $
$\therefore \, \tan ^{2}\, \theta =\frac{144}{25}$
$\Rightarrow \, \sec ^{2} \theta-1= \frac{144}{25}$
$ \Rightarrow \,\sec ^{2} \,\theta=\frac{169}{25} $
$ \therefore \, \sec \,\theta =\frac{13}{5}$