For $x\in \left(0,\frac{\pi }{2}\right),$ if $\mathrm{c}\mathrm{o}{\mathrm{s}}^{-1}\left(\frac{7}{2}\left(1+\cos 2x\right)+\sqrt{\left(\mathrm{s}\mathrm{i}{\mathrm{n}}^2 x-48\mathrm{c}\mathrm{o}{\mathrm{s}}^2 x\right)}\sin x\right)$$=x-{\cos }^{-1}\left(k\cos x\right),$ 
then the value of $k$ is equal to

• A

  $1$
• B

  $5$
• Correct answer
  C

  $7$
• D

  $14$

Hence, the correct answer is option: 'C'.