Question

Prove the identity. $$\cos x-\cos (3 x)=4 \sin ^{2} x \cos x$$

   Prove the identity.
$$\cos x-\cos (3 x)=4 \sin ^{2} x \cos x$$

Precalculus

Jay Abramson 1st Edition

Chapter 7, Problem 222

Instant Answer

Solved by Expert Alisa Lu

07/23/2019

Step 1

The sum-to-product formula is given by $\cos(\alpha) - \cos(\beta) = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right)$. Here, $\alpha = x$ and $\beta = 3x$. So, we get $$\cos(x) - \cos(3x) = -2\sin\left(\frac{x + Show more…

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Prove the identity. $$\cos x-\cos (3 x)=4 \sin ^{2} x \cos x$$

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Key Concepts

Double Angle Identities

Double angle identities, such as sin 2x = 2 sin x cos x, are used to simplify products of sine and cosine functions into single trigonometric functions. This type of simplification is often a key step in rearranging and proving more complex trigonometric identities.

Sum-to-Product Formulas

These formulas allow the conversion of sums or differences of trigonometric functions into products, which simplifies the process of proving identities. In this context, the formula cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2) transforms the difference of cosines into a product of sines, making it easier to factor and compare terms.

Properties of Even and Odd Functions

Trigonometric functions have symmetry properties that are useful in simplifying expressions. Cosine is an even function, meaning cos(-x) = cos x, while sine is an odd function, meaning sin(-x) = -sin x. These properties help in handling negative angles when applying the sum-to-product identities.

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