Question

If $xϵ(0,\frac{\pi }{2})$, then
${\cos }^{-1}(\frac{7}{2}(1+\cos 2x)+\sqrt{({\sin }^{2}x-48{\cos }^{2}x)}\sin x)=ax-{\cos }^{-1}\left(b\cos x\right)$.
Find the value of $a+b$

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

The correct option is C 8

Let $y={\cos }^{-1}(\frac{7}{2}(1+\cos 2x)+\sqrt{({\sin }^{2}x-48{\cos }^{2}x)}\sin x)$

$={\cos }^{-1}(\left(7\cos x\right)\left(\cos x\right)+\sqrt{1-49{\cos }^{2}x}\sqrt{1-{\cos }^{2}x})......co{s}^{-1}x+co{s}^{-1}yformula$

$={\cos }^{-1}\left(\cos x\right)-{\cos }^{-1}\left(7\cos x\right)$ $=x-{\cos }^{-1}\left(7\cos x\right)$

$\Rightarrow a+b=7+1=8$