Integration - Integrable Forms of Trigonometric Functions

When the integrand involves even powers of sines or cosines, we recall the Double Angle Formulae for cosine:

\[ \begin{align*} \cos^2 \theta & = \frac{1}{2} (1 + \cos 2\theta) \\ & = \frac{1}{2} (1 - \cos 2\theta). \end{align*} \]

Proof.

Recall that:

\[ \cos 2\theta = 2 \cos^2 \theta - 1 \]

Rearranging, we have
\[ \begin{align*} 2 \cos^2 \theta & = 1 + \cos 2\theta \\ \cos^2 \theta & = \frac{1}{2} (1 + 2 \cos 2\theta). \end{align*} \]

Similarly, for sine,
\[ \cos 2\theta = 1 - 2 \sin^2 \theta. \]

So, we have
\[ \begin{align*} 2 \sin^2 \theta & = 1 - \cos 2\theta \\ \sin^2 \theta & = \frac{1}{2} (1- \cos 2\theta). \end{align*} \]

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