Question

If $\sec \theta =2x$ and $y\tan \theta =2$, then the value of $2[x^2-\frac{1}{y^2}]$ is

• A. $1$
• B. $\frac{1}{2}$
• C. $\frac{1}{3}$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{2}$

$\sec \theta =2x$ $,$ $y\tan \theta =2$

$\Rightarrow {\sec }^2\theta =4x^2\text{and}{y}^2{\tan }^2\theta =4$

$\Rightarrow {\sec }^2\theta =2\cdot 2x^2\text{and}{\tan }^2\theta =\frac{4}{y^2}=2\cdot \frac{2}{y^2}$

$\Rightarrow 2x^2=\frac{1}{2}{\sec }^2\theta \text{and}\frac{2}{y^2}=\frac{1}{2}{\tan }^2\theta$

$\Rightarrow$ Now,

$2[x^2-\frac{1}{y^2}]=2x^2-\frac{2}{y^2}$

$=\frac{1}{2}{\sec }^2\theta -\frac{1}{2}{\tan }^2\theta$

$=\frac{1}{2}[{\sec }^2\theta -{\tan }^2\theta ]$

$=\frac{1}{2}\cdot 1$

$=\frac{1}{2}[\therefore {\sec }^2\theta -{\tan }^2\theta =1]$

Hence, the answer is $\frac{1}{2}.$