Answer
$$0$$
Work Step by Step
Given $$\int _{-\pi/4}^{\pi/4}\frac{t^4\tan t}{2+\cos t}dt$$
Since $f(t)=\frac{t^4\tan t}{2+\cos t}$ is an odd function and continuous on $[-\pi/4,\pi/4]$, then
$$\int _{-\pi/4}^{\pi/4}\frac{t^4\tan t}{2+\cos t}dt=0$$