# Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Sine

## Theorem

$\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$

## Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

 $\displaystyle u$ $=$ $\displaystyle \frac {\sin^{m - 1} a x} {\cos^n a x}$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle \frac {\cos^n a x \dfrac {\d} {\d x} \sin^{m - 1} a x - \sin^{m - 1} a x \dfrac {\d} {\d x} \cos^n a x} {\cos^{2 n} a x}$ Quotient Rule for Derivatives

Thus:

 $\displaystyle \frac \d {\d x} \sin^{m - 1} a x$ $=$ $\displaystyle a \paren {m - 1} \sin^{m - 2} a x \cos a x$ Derivative of $\sin a x$, Derivative of Power, Chain Rule for Derivatives $\displaystyle \frac \d {\d x} \cos^n a x$ $=$ $\displaystyle a n \cos^{n - 1} a x \paren {-\sin a x}$ Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle -a n \cos^{n - 1} a x \sin a x$

and so:

 $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle \frac {\cos^n a x \paren {a \paren {m - 1} \sin^{m - 2} a x \cos a x} + \sin^{m - 1} a x \paren {a n \cos^{n - 1} a x \sin a x} } {\cos^{2 n} a x}$ $\displaystyle$ $=$ $\displaystyle \frac {a \paren {m - 1} \sin^{m - 2} a x \cos^2 a x + a n \sin^m a x} {\cos^{n + 1} a x}$ simplifying and cancelling $\cos^{n - 1}$ $\displaystyle$ $=$ $\displaystyle \frac {a \sin^{m - 2} a x \paren {\paren {m - 1} \cos^2 a x + n \sin^2 a x} } {\cos^{n + 1} a x}$ factorising $\displaystyle$ $=$ $\displaystyle \frac {a \sin^{m - 2} a x \paren {\paren {m - 1} \paren {1 - \sin^2 a x} + n \sin^2 a x} } {\cos^{n + 1} a x}$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle \frac {a \sin^{m - 2} a x \paren {\paren {n - m + 1} \sin^2 a x + \paren {m - 1} } } {\cos^{n + 1} a x}$ simplifying $\displaystyle$ $=$ $\displaystyle \frac {a \paren {n - m + 1} \sin^m a x} {\cos^{n + 1} a x} + \frac {a \paren {m - 1} \sin^{m - 2} a x} {\cos^{n + 1} a x}$ separating

Then let:

 $\displaystyle \frac {\d v} {\d x}$ $=$ $\displaystyle \sin a x$ $\displaystyle \leadsto \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {-\cos a x} a$ Primitive of $\sin a x$

Then:

 $\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \rd x$ $=$ $\displaystyle \int \frac {\sin^{m - 1} a x} {\cos^n a x} \sin^a x \rd x$ $\displaystyle$ $=$ $\displaystyle \paren {\frac {\sin^{m - 1} a x} {\cos^n a x} } \paren {\frac {-\cos a x} a}$ Integration by Parts $\displaystyle$  $\, \displaystyle - \,$ $\displaystyle \int \paren {\frac {-\cos a x} a} \paren {\frac {a \paren {n - m + 1} \sin^m a x} {\cos^{n + 1} a x} + \frac {a \paren {m - 1} \sin^{m - 2} a x} {\cos^{n + 1} a x} } \rd x + C$ $\displaystyle$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x}$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \paren {n - m + 1} \int \frac {\sin^m a x} {\cos^n a x} \rd x$ Linear Combination of Integrals $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$
 $\displaystyle \leadsto \ \$ $\displaystyle \paren {1 - \paren {n - m + 1} } \int \frac {\sin^m a x} {\cos^n a x} \rd x$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x} + \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {m - n} \int \frac {\sin^m a x} {\cos^n a x} \rd x$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x} {a \cos^{n - 1} a x} + \paren {m - 1} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$ $\displaystyle \leadsto \ \$ $\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \rd x$ $=$ $\displaystyle \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$

$\blacksquare$