Answer
$$\int_{0}^{\frac{\pi}{4}}\tan ^{3}\theta\sec ^{2}\theta d\theta=\frac{1}{4}$$
Work Step by Step
$$\int_{0}^{\frac{\pi}{4}}\tan ^{3}\theta\sec ^{2}\theta d\theta=\int_{0}^{\frac{\pi}{4}}\tan ^{3}\theta d\tan \theta$$
$$=\left [ \frac{\tan ^{4} \theta}{4} \right ]_{0}^{\frac{\pi}{4}}$$
$$=\frac{1}{4}$$