Problem 69 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\cos^4 x - \sin^4 x = \cos 2x\) by using the difference of squares and double angle identity.

Step 1: Recognize the Formula

The expression given is a difference of squares: \(\cos^4 x - \sin^4 x\). Utilize the identity for the difference of squares, \(a^2 - b^2 = (a + b)(a - b)\).

Step 2: Apply the Difference of Squares Formula

Apply the formula to rewrite \(\cos^4 x - \sin^4 x\) as \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\).

Step 3: Simplify the Expression

Recall the Pythagorean identity \(\cos^2 x + \sin^2 x = 1\). Substitute this identity into the expression to simplify it to \(1(\cos^2 x - \sin^2 x)\).

Step 4: Use Double Angle Identity

Recognize that \(\cos^2 x - \sin^2 x\) is equivalent to the double angle identity for cosine, \(\cos 2x\).

Step 5: Write the Final Simplified Expression

Thus, the expression \(\cos^4 x - \sin^4 x\) simplifies to \(\cos 2x\), proving the identity.

Key Concepts

Difference of squares

The difference of squares is a fundamental algebraic concept frequently used to simplify expressions. It refers to an equation of the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). This makes solving expressions more straightforward by breaking them into simpler parts.

• In the expression \(\cos^4 x - \sin^4 x\), we see it follows the pattern of a difference of squares.
• Here, \(a\) is \(\cos^2 x\) and \(b\) is \(\sin^2 x\).

So, applying the difference of squares, \(\cos^4 x - \sin^4 x\) becomes \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\). This manipulation simplifies our original equation into a more manageable form, setting the stage for further simplifications.

Pythagorean identity

The Pythagorean identity is a key trigonometric identity that states \(\cos^2 x + \sin^2 x = 1\). This simple yet powerful relationship helps us manage equations involving trigonometric functions. Here's how it assists in our problem:

• After the first application of the difference of squares, we have the expression \((\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\).
• Using the Pythagorean identity, the term \(\cos^2 x + \sin^2 x\) simplifies to 1.

This reduction further simplifies our equation to \(1(\cos^2 x - \sin^2 x)\), effectively leaving us with just \(\cos^2 x - \sin^2 x\). Our equation becomes easier to handle and sets the stage for applying additional identities.

Double angle identity

The double angle identity for cosine is a trigonometric formula that expresses \(\cos 2x\) in terms of \(\cos x\) and \(\sin x\). One version of this identity states that \(\cos 2x = \cos^2 x - \sin^2 x\).

• In the previous step, we derived the expression \(\cos^2 x - \sin^2 x\).
• This matches perfectly with the double angle identity for cosine.

By recognizing this, the expression \(\cos^2 x - \sin^2 x\) can now be rewritten simply as \(\cos 2x\). This transformation confirms the original equation \(\cos^4 x - \sin^4 x = \cos 2x\). The use of trigonometric identities, like the double angle identity, allows us to verify and simplify expressions involving trigonometric functions effectively.