Question

If $\text{cosec}\theta -\cot \theta =\frac{1}{2},$ then the value of ${\sec }^2\theta -{\cos }^2\theta$ is equal to

• A. $\frac{44}{625}$
• B. $\frac{616}{225}$
• C. $\frac{706}{625}$
• D. $\frac{544}{225}$

Answer 1

The correct option is D $\frac{544}{225}$

$\text{cosec}\theta -\cot \theta =\frac{1}{2}...\left(1\right)$
We know that, ${\text{cosec}}^2\theta -{\cot }^2\theta =1$
$\Rightarrow (\text{cosec}\theta +\cot \theta )(\text{cosec}\theta -\cot \theta )=1$
$\Rightarrow \text{cosec}\theta +\cot \theta =2...\left(2\right)$

From equation $\left(1\right)$ and $\left(2\right)$
$\text{cosec}\theta =\frac{5}{4}\Rightarrow {\sin }^2\theta =\frac{16}{25}\Rightarrow {\cos }^2\theta =\frac{9}{25}$

Now,
${\sec }^2\theta -{\cos }^2\theta =\frac{25}{9}-\frac{9}{25}=\frac{544}{225}$