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)

$\frac{13}{5}$

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

$\Rightarrow {\sec }^2\theta =\frac{169}{25}$
$\Rightarrow \sec \theta =\frac{13}{5}$