Question

$\mathrm{If}\mathrm{tan\theta }+\mathrm{sec\theta }=\frac{3}{2},\mathrm{then}\mathrm{tan\theta }\mathrm{is}$

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

Answer 1

The correct option is B

$\frac{5}{12}$
Given, $\mathrm{sec\theta }+\mathrm{tan\theta }=\frac{3}{2}$
Now, we know the trigonometric identity: ${\sec }^2\theta -{\tan }^2\theta =1$.
Then, ${\sec }^2\theta -{\tan }^2\theta =1\Rightarrow (\mathrm{sec\theta }-\mathrm{tan\theta })(\mathrm{sec\theta }+\mathrm{tan\theta })=1\Rightarrow (\mathrm{sec\theta }-\mathrm{tan\theta })\times \frac{3}{2}=1\Rightarrow (\mathrm{sec\theta }-\mathrm{tan\theta })=\frac{2}{3}$
Now, subtracting both equations we get,
$(\mathrm{sec\theta }+\mathrm{tan\theta })-(\mathrm{sec\theta }-\mathrm{tan\theta })=\frac{3}{2}-\frac{2}{3}\Rightarrow 2\mathrm{tan\theta }=\frac{5}{6}\Rightarrow \mathrm{tan\theta }=\frac{5}{12}$
Hence, the option b. is correct.