Answer
$$\left\langle {0,1} \right\rangle $$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\left\langle {\cos 2t,\sin 2t} \right\rangle } dt \cr
& {\text{Integrate}} \cr
& \int_0^{\pi /2} {\left\langle {\cos 2t,\sin 2t} \right\rangle } dt = \left[ {\left\langle {\frac{1}{2}\sin 2t, - \frac{1}{2}\cos 2t} \right\rangle } \right]_0^{\pi /2} \cr
& {\text{Evaluate the limits of integration}} \cr
& = \left\langle {\frac{1}{2}\sin 2\left( {\frac{\pi }{2}} \right) - \frac{1}{2}\sin 2\left( 0 \right), - \frac{1}{2}\cos 2\left( {\frac{\pi }{2}} \right) + \frac{1}{2}\cos 2\left( 0 \right)} \right\rangle \cr
& {\text{Simplifying}} \cr
& = \left\langle {\frac{1}{2}\sin \pi - \frac{1}{2}\sin 0, - \frac{1}{2}\cos \pi + \frac{1}{2}\cos 0} \right\rangle \cr
& = \left\langle {0, - \frac{1}{2}\left( { - 1} \right) + \frac{1}{2}\left( 1 \right)} \right\rangle \cr
& = \left\langle {0,1} \right\rangle \cr} $$
