Question

If $\cos \left\{{\tan }^{-1}\left(\sin \left({\cot }^{-1}x\right)\right)\right\}={\left(\frac{x^a+1}{x^b+2}\right)}^{1/c}$. Find the value of $a+b+c$

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is D $6$

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

$=\cos \left\{{\tan }^{-1}\frac{1}{\sqrt{1+x^2}}\right\}=\cos \left\{{\cos }^{-1}\frac{\sqrt{1+x^2}}{\sqrt{2+x^2}}\right\}={\left(\frac{1+x^2}{2+x^2}\right)}^{1/2}$

$\therefore a+b+c=2+2+2=6$