These formulas express powers of sine, cosine, and tangent functions in terms of trigonometric functions of multiple angles. They help simplify expressions involving higher powers of trigonometric functions.
\[ \sin^2 \alpha = \frac{1}{2}(1 - \cos 2\alpha) \]
Explanation: The square of sine can be rewritten using the double-angle cosine function. This is useful for integrating or simplifying trigonometric expressions.
\[ \cos^2 \alpha = \frac{1}{2}(1 + \cos 2\alpha) \]
Explanation: Similarly, cosine squared is expressed using the double-angle cosine, facilitating simplification.
\[ \tan^2 \alpha = \frac{1 - \cos 2\alpha}{1 + \cos 2\alpha} \]
Explanation: Tangent squared is given as a ratio involving cosine double-angle, connecting tangent to cosine functions.
\[ \sin^3 \alpha = \frac{1}{4}(3 \sin \alpha - \sin 3\alpha) \]
Explanation: The cube of sine can be broken down into a linear combination of sine of \(\alpha\) and sine of triple angle \(3\alpha\).
\[ \cos^3 \alpha = \frac{1}{4}(3 \cos \alpha + \cos 3\alpha) \]
Explanation: Similar to sine, cosine cubed is expressed as a sum of cosine terms with different angles.
\[ \sin^4 \alpha = \frac{1}{8}(\cos 4\alpha - 4 \cos 2\alpha + 3) \]
Explanation: Fourth power of sine is written as a combination of cosine functions with multiple angles, aiding in integration and simplification.
\[ \cos^4 \alpha = \frac{1}{8}(\cos 4\alpha + 4 \cos 2\alpha + 3) \]
Explanation: Fourth power of cosine follows a similar pattern with different signs.
\[ \sin^5 \alpha = \frac{1}{16}(\sin 5\alpha - 5 \sin 3\alpha + 10 \sin \alpha) \]
Explanation: The fifth power of sine is expressed as a sum of sines of multiple angles, useful in series expansions and signal processing.
\[ \cos^5 \alpha = \frac{1}{16}(\cos 5\alpha + 5 \cos 3\alpha + 10 \cos \alpha) \]
Explanation: Similarly, fifth power of cosine is decomposed into cosines of multiple angles.