Question

If $\sec \theta -\tan \theta =\frac{1}{2},$ then $\theta$ lies in

• A. the first quadrant
• B. the second quadrant
• C. the third quadrant
• D. the fourth quadrant

Answer 1

Answer: (A)

Given $\sec \theta -\tan \theta =\frac{1}{2}$ ....(1)
Also ${\sec }^2\theta -{\tan }^2\theta =1$ ....(2)
Dividing (2) by (1), we get
$\sec \theta +\tan \theta =2$ ....(3)
Adding (1) and (3), we get
$2\sec \theta =\frac{5}{2}\Rightarrow \sec \theta =\frac{5}{4}$
and subtracting (2) from (1), we get
$2\tan \theta =\frac{3}{2}\Rightarrow \tan \theta =\frac{3}{4}$
Since both sec $\theta$ and tan $\theta$ are positive, therefore, $\theta$ lies in the first quadrant.