Question

If $\text{cosec}\theta =2x$ and $y\cot \theta =2$, then the value of $4(x^2-\frac{1}{y^2})$ is :

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

Answer 1

The correct option is A $1$

$\text{cosec}\theta =2x$

Squaring both sides:

${\text{cosec}}^2\theta =4x^2$

$x^2=\frac{1}{4}{\text{cosec}}^2\theta \dots \left(1\right)$

$y\cot \theta =2$

Squaring both sides:

$y^2{\cot }^2\theta =4$

$\frac{1}{y^2}=\frac{1}{4}{\cot }^2\theta \dots \left(2\right)$

Subtract $\left(2\right)$ from $\left(1\right)$:

$4(x^2-\frac{1}{y^2})=4[\frac{1}{4}{\text{cosec}}^2\theta -\frac{1}{4}{\cot }^2\theta ]$

$=4\times \frac{1}{4}[{\text{cosec}}^2\theta -{\cot }^2\theta ]$

$=1.1$

$=1$

Hence, the answer is $1.$