Problem 68 - Algebra and Trigonometry

Question

59–76 Prove the identity. $$4\left(\sin ^{6} x+\cos ^{6} x\right)=4-3 \sin ^{2} 2 x$$

Step-by-Step Solution

Answer

The identity is proven by simplifying both sides to show equality using trigonometric identities and algebraic factoring.

Step 1: Understand the Identity

We need to prove that \( 4(\sin^6 x + \cos^6 x ) = 4 - 3 \sin^2 2x \). This means showing both sides of the equation are equal.

Step 2: Use Algebraic Identity

Notice that \( \sin^6 x + \cos^6 x \) can be rewritten using \((a+b)^3 = a^3 + b^3 \) and \(\sin^2 x + \cos^2 x = 1\). Specifically, \( \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) = 1 - 3\sin^2 x \cos^2 x \).

Step 3: Transform Sin^2 2x

Recall that \( \sin^2 2x = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x \). Substitute this into the equation: \( -3 \sin^2 2x = -12 \sin^2 x \cos^2 x \).

Step 4: Substitute and Simplify

Substitute \( 1 - 3\sin^2 x \cos^2 x \) for \( \sin^6 x + \cos^6 x \) in the original expression: \(4(1 - 3\sin^2 x \cos^2 x) = 4 - 12\sin^2 x \cos^2 x \). This matches \( 4 - 3\sin^2 2x \), showing both sides are equal.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a powerful mathematical tool used to simplify expressions and solve equations. In the given trigonometric identity problem, algebraic manipulation helps us reframe the expressions for easier comparison. For instance, we use the basic identity

• \( (a+b)^3 = a^3 + b^3 \)
• And the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \).

By using these identities, we can rewrite \( \sin^6 x + \cos^6 x \) as \((\sin^2 x + \cos^2 x)(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x),\) which simplifies to \(1 - 3\sin^2 x \cos^2 x.\)
This step shows the strength of algebraic manipulation in breaking down complex expressions into manageable pieces. It allows us to clearly see how parts of the equation can relate or cancel out when simplifying trigonometric identities.

Sine and Cosine Functions

Sine and cosine are fundamental to trigonometry, each describing the ratio of specific sides of a right triangle relative to an angle. In this problem, understanding these functions helps us simplify and transform expressions. Known identities such as

• \( \sin^2 x + \cos^2 x = 1 \)

are central in transforming more complex sine and cosine expressions. Here, our given identity needs rewriting \( \sin^6 x \) and \( \cos^6 x \) using \( \sin^2 x \) and \( \cos^2 x \).
These trigonometric concepts not only provide shortcuts for algebraic manipulation but also serve as steps toward understanding more complex derivative trigonometric formulas. By mastering sine and cosine functions, we can tackle a wide range of trigonometric problems with confidence.

Double Angle Formula

The Double Angle Formula is a key trigonometric identity that allows us to express trigonometric functions of doubled angles in terms of single angles. Specifically, it is used frequently when working with expressions like \( \sin 2x \) and \( \cos 2x \). In this exercise, we focus on the sine double angle formula:

• \( \sin 2x = 2 \sin x \cos x \)

To simplify our identity, we also consider

• \( \sin^2 2x = (2 \sin x \cos x)^2 = 4 \sin^2 x \cos^2 x. \)

This substitution into the identity \( 4 - 3 \sin^2 2x \) gives us \( 4 - 12 \sin^2 x \cos^2 x, \)which matches the transformation on the left-hand side: \( 4(1 - 3 \sin^2 x \cos^2 x). \)
The Double Angle Formula not only simplifies complex expressions but also reveals deeper relationships within trigonometric functions. Mastery of this concept is crucial for advancing in trigonometry and integrating these functions into other mathematical areas.