Answer
$$\frac{{{{\tan }^5}t}}{5} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{{\sin }^4}t}}{{{{\cos }^6}t}}} dt \cr
& {\text{Write the integrand as}} \cr
& {\text{ = }}\int {\frac{{{{\sin }^4}t}}{{{{\cos }^4}t{{\cos }^2}t}}dt} \cr
& {\text{ = }}\int {\left( {\frac{{{{\sin }^4}t}}{{{{\cos }^4}t}}} \right)\left( {\frac{1}{{{{\cos }^2}t}}} \right)dt} \cr
& {\text{Use trigonometric identities}} \cr
& {\text{ = }}\int {{{\tan }^4}t{{\sec }^2}tdt} \cr
& {\text{Integrate by using the power rule for integration}} \cr
& = \frac{{{{\tan }^5}t}}{5} + C \cr} $$
