Verified Answer :

\(\int\limits^ \frac{ \pi }{4} _0 {cos2t} \, dt = \frac{sin2t}{2} I\limits^ \frac{ \pi }{4} _0 = \\ = \frac{sin( \pi/2) }{2}- \frac{sin (0)}{2}= \frac{1}{2}\)
Answer: 1/2