I did this trigonometric integral in two different ways, and the results that I got were with two different trigonometric functions, $\sec x$ and $\tan x$.
The integral is: $\mathbf{\int tan^{5}x \, \, \, sec^{4}x \; \; dx} \\ $
Result 1: $\frac{\tan^{8}x}{8} + \frac{\tan^{6}x}{6} + C \\$
Result 2: $\frac{\sec^{8}x}{8} + \frac{1}{3} \sec^{6}x + \frac{\sec^{4}x}{4} + C$
My questions is: ¿How do I prove that these results are the same?
Thanks in advance.