Question

The value of $\left({\mathrm{cosec}}^2\left({\cot }^{-1}x\right)-{\cot }^2\left({\mathrm{cosec}}^{-1}x\right)\right)$ is equal to

• A. $2\left(1+x^2\right)$
• B. 2
• C. $2x$
• D. $2x^2$

Answer 1

Answer: (B)

As, $1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta$
$\text{ so, }{\mathrm{cosec}}^2\left({\cot }^{-1}x\right)=1+{\cot }^2\left({\cot }^{-1}x\right)$
$\text{ and }{\cot }^2\left({\mathrm{cosec}}^{-1}x\right)={\mathrm{cosec}}^2\left({\mathrm{cosec}}^{-1}x\right)-1=\left(x^2-1\right)$
$\text{ so, }{\mathrm{cosec}}^2\left({\cot }^{-1}x\right)-{\cot }^2\left({\mathrm{cosec}}^{-1}x\right)=\left(1+x^2\right)$
$\text{ so, }{\mathrm{cosec}}^2\left({\cot }^{-1}x\right)-{\cot }^2\left({\mathrm{cosec}}^{-1}x\right)=\left(1+x^2\right)-\left(x^2-1\right)=1+1=2$