In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x

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Recognize that the expression involves a power of sine: \(\sin^4 x\). Our goal is to rewrite \(\sin^4 x\) using power-reducing formulas so that the expression contains only first powers of trigonometric functions.

Recall the power-reducing formula for \(\sin^2 x\): \(\sin^2 x = \frac{1 - \cos(2x)}{2}\)

Express \(\sin^4 x\) as \((\sin^2 x)^2\) and substitute the power-reducing formula for \(\sin^2 x\): \(\sin^4 x = \left( \frac{1 - \cos(2x)}{2} \right)^2\)

Expand the square to get an expression in terms of \(\cos(2x)\): \(\sin^4 x = \frac{(1 - \cos(2x))^2}{4} = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}\)

Apply the power-reducing formula again to \(\cos^2(2x)\): \(\cos^2(2x) = \frac{1 + \cos(4x)}{2}\), then substitute this back into the expression to write \(\sin^4 x\) entirely in terms of cosines with powers of 1.

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Power-Reducing Formulas

Power-reducing formulas express powers of sine and cosine functions in terms of first powers of trigonometric functions with multiple angles. For example, sin²x can be rewritten as (1 - cos 2x)/2. These formulas simplify expressions by reducing the exponent, making integration or further manipulation easier.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. They allow rewriting expressions in different but equivalent forms. Understanding identities like double-angle and half-angle formulas is essential to apply power-reducing formulas correctly.

Exponentiation of Trigonometric Functions

Raising trigonometric functions to powers, such as sin⁴x, involves repeated multiplication. To simplify or integrate such expressions, it is necessary to rewrite them using identities that reduce the power, often by expressing higher powers in terms of lower powers or multiple angles.