If g(x)=∫limits_0^x cos ^4 t d t, then g(x+π)=

$g(x+\pi)=\int\limits_0^{\pi+x} \cos ^4 t d t$

$=\int\limits_0^\pi \cos ^4 t d t+\int\limits_\pi^{\pi+x} \cos ^4 x d x$

(Put $t=\pi+\theta$ is second integral)

$=\int\limits_0^\pi \cos ^4 t d t+\int\limits_0^x \cos ^4 t d t=g(\pi)+g(x)$

Hence (1) is the correct answer.