If $\underset{0}{\overset{\frac{\pi }{2 }}{\int }}\log \cos x dx=\frac{\pi }{2}\log \left(\frac{1}{2}\right),$ then $\underset{0}{\overset{\frac{\pi }{2 }}{\int }}\log \sec x dx=$

• A

  $\frac{\pi }{2}\log \left(\frac{1}{2}\right)$
• B

  $1-\frac{\pi }{2}\log \left(\frac{1}{2}\right)$
• C

  $1+\frac{\pi }{2}\log \left(\frac{1}{2}\right)$
• Correct answer
  D

  $\frac{\pi }{2}\log 2$

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