Question

For $x\in \left(0 , \frac{\pi }{2}\right),$ if $cos^{- 1}\left(\frac{7}{2} \left(1 + cos 2 x\right) + \sqrt{\left(s i n^{2} \, x - 48 c o s^{2} \, x\right)} sin ⁡ x\right)$ $=x-\left(cos\right)^{- 1}\left(k cos x\right),$

then the value of $k$ is equal to

• A. $1$
• B. $5$
• C. $7$
• D. $14$

Answer 1

Answer: (C)

$y=cos^{- 1}\left(\frac{7}{2} \left(1 + cos 2 x\right) + \sqrt{\left(s i n^{2} \, x - 48 c o s^{2} \, x\right)} sin ⁡ x\right)$
$=cos^{- 1}\left(\left(7 cos x\right) \left(cos ⁡ x\right) + \sqrt{1 - 49 \left(cos\right)^{2} x} \sqrt{1 - \left(cos\right)^{2} x}\right)$
$=cos^{- 1}\left(cos x\right)-cos^{- 1}\left(7 cos ⁡ x\right)$
$=x-cos^{- 1}\left(7 cos x\right)$
Hence, the value of $k=7$