Question

Assertion :$tan[co{s}^{-1}\left(\frac{1}{\sqrt{82}}\right)-si{n}^{-1}\left(\frac{5}{\sqrt{26}}\right)]$ is equal to $\frac{29}{3}$. Reason: $\left\{xcos\right(co{t}^{-1}x)+sin(co{t}^{-1}x){\}}^2=\frac{51}{50}$ if $x=\frac{1}{5\sqrt{2}}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Reason:
Let ${\cot }^{-1}x=y\Rightarrow x=\cot y$

$\left(x\cos \right({\cot }^{-1}x\left)\right)+\sin {\left({\cot }^{-1}x\right)}^2={(\cot y\cos y+\sin y)}^2={\left(\frac{{\cos }^{2}x+{\sin }^{2}x}{\sin x}\right)}^2$

$=cose{c}^{2}y=1+co{t}^{2}y=1+x^2$

For $x=\frac{1}{5\sqrt{2}}$
$\left(xcosco{t}^{-1}x\right)+]sin{\left(co{t}^{-1}x\right)}^2=1+\frac{1}{(5\sqrt{2}{)}^2}=1+\frac{1}{50}=\frac{51}{50}$
Assertion:
$\tan ({\cos }^{-1}\left(\frac{1}{\sqrt{82}}\right)-{\sin }^{-1}\left(\frac{5}{\sqrt{26}}\right))=\tan ({\tan }^{-1}9-{\tan }^{-1}5)$

$=\tan \left({\tan }^{-1}\frac{2}{23}\right)=\frac{2}{23}\ne \frac{29}{3}$

Hence assertion is wrong and reason is correct