There's a trick to integrating the product of a higher-degree tangent function and a higher-degree secant function. This is an example in which the power of the secant function is even. To use this method, the power of the tangent function can be even or odd.

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Tan^m sec^n^ even n^ Example 2: Calculus-Integrals: Trigonometric Integrals --(5:16).