Problem 63 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$\frac{\sin 4 x}{\sin x}=4 \cos x \cos 2 x$$

Step-by-Step Solution

Answer

The identity is proven: \(\frac{\sin 4 x}{\sin x}=4 \cos x \cos 2 x\).

Step 1: Express sin(4x) using angle sum identities

We start by expressing \(\sin 4x\) as \(\sin(2 \times 2x)\). Using the double angle identity, \(\sin(2A) = 2 \sin A \cos A\), we can write \(\sin 4x = \sin(2 \times 2x) = 2 \sin 2x \cos 2x\). We know that \(\sin 2x = 2 \sin x \cos x\). Substitute this into the expression: \(\sin 4x = 2(2 \sin x \cos x) \cos 2x = 4 \sin x \cos x \cos 2x\).

Step 2: Simplify the left side of the identity

The left side of the identity is \(\frac{\sin 4x}{\sin x}\). Substitute \(\sin 4x\) from Step 1: \(\frac{4 \sin x \cos x \cos 2x}{\sin x}\).

Step 3: Cancel common terms

Now, we cancel \(\sin x\) from the numerator and denominator: \(\frac{4 \sin x \cos x \cos 2x}{\sin x} = 4 \cos x \cos 2x\).

Step 4: Verify the result

After cancellation, we have shown that the left side simplifies to \(4 \cos x \cos 2x\), which matches the right side of the identity. This verifies the identity.

Key Concepts

Angle Sum Identities

To truly grasp trigonometric identities, one vital tool in your toolkit is the angle sum identities. These identities help us express the trigonometric functions of sums or differences of angles through simpler functions of individual angles.
For example, the identity for sine with two angles, commonly written as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), allows us to break down complex angle expressions.

Learning to use these identities gives you a powerful way to simplify expressions or prove identities, as seen in the exercise where the expression \( \sin 4x \) is broken down. By expressing \( 4x \) as \( 2 \times 2x \) and applying the angle sum formula, we simplify the trigonometric components involved.
Angle sum identities are not only concerned with sine and cosine but also cover tangent. For instance, \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \). Knowing these identities by heart will considerably ease your work with trigonometric problems.

• They help transform complex angle expressions.
• Aid in proving trigonometric identities.
• Make equations more manageable.

Being comfortable with these identities means you can confidently approach a wide array of mathematical challenges.

Double Angle Identity

Another key concept used in trigonometry is the double angle identities. These identities simplify expressions where an angle is doubled, such as \( \sin(2x), \cos(2x), \text{and} \tan(2x) \).
The double angle identities for sine and cosine are:
- \( \sin(2x) = 2\sin x \cos x \)
- \( \cos(2x) = \cos^2 x - \sin^2 x \)
These formulas allow you to rewrite expressions with doubled angles regarding single angles, making them more manageable for simplification.


In our exercise, the double angle identity was crucial in expressing \( \sin(4x) \) as a product of simpler trigonometric functions. By understanding and applying these identities, we were able to bridge the gap between complex trigonometric formulas and simpler expressions, leading us to solve the identity.

• Simplify complex trigonometric expressions.
• Provide a bridge to simplify or prove further identities.
• Help in converting between different trigonometric forms.

Mastering the double angle identities equips you to handle a diverse range of trigonometric problems.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is often key to solving trigonometric identities. It involves reducing complex expressions to simpler, equivalent ones.
A central part of simplification is identifying and canceling common terms. As shown in our exercise, we reduced \( \frac{\sin 4x}{\sin x} \) to \( 4 \cos x \cos 2x \).

The process typically entails converting expressions using known identities, like angle sum or double angle identities, and simplifying ratios and products by canceling out terms that appear in both the numerator and the denominator.

• Begin by expressing larger angles in terms of smaller ones using angle identities.
• Look for opportunities to factor out common terms.
• Simplify fractions by canceling common terms.

This step-by-step simplification not only solves equations but also strengthens understanding. Once you achieve a thorough comprehension, handling similar problems becomes intuitive and rewarding.