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Chapter 20: Problem 50

Solve the given problems. Find the exact value of \(\cos ^{4} x-\sin ^{4} x-\cos 2 x\).

Short Answer

Expert verified
The exact value of the expression is 0.

Step by step solution

01

Recognizing a Pattern

Notice that the expression \( \cos^4 x - \sin^4 x \) can be expressed using an algebraic identity for the difference of squares. Recall the identity: \( a^2 - b^2 = (a-b)(a+b) \). In this context, we can use \( a = \cos^2 x \) and \( b = \sin^2 x \).
02

Applying the Difference of Squares

Apply the identity: \[ \cos^4 x - \sin^4 x = (\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x) \]. Since \( \cos^2 x + \sin^2 x = 1 \) due to the Pythagorean identity, we have: \[ \cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x. \]
03

Substituting Cosine Double Angle Identity

We know that \( \cos 2x = \cos^2 x - \sin^2 x \). Thus, the problem simplifies to \[ \cos^2 x - \sin^2 x - \cos 2x. \] Substitute for \( \cos 2x \): \[ \cos^2 x - \sin^2 x - (\cos^2 x - \sin^2 x) = 0. \]
04

Conclusion

Since the expression simplifies to \( 0 \), the exact value of the original expression \( \cos^4 x - \sin^4 x - \cos 2x \) is \( 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a powerful algebraic identity that simplifies expressions involving squares of different terms. The identity states:
  • \(a^2 - b^2 = (a-b)(a+b)\)
Using this identity, we can break down complex expressions into simpler factors. For instance, when we have \(\cos^4 x - \sin^4 x\), we equate \(a\) to \(\cos^2 x\) and \(b\) to \(\sin^2 x\). This transforms the expression into:
  • \((\cos^2 x - \sin^2 x)(\cos^2 x + \sin^2 x)\)
By applying this identity, it simplifies our calculations and sets the stage for further simplification using other identities.
Pythagorean Identity
The Pythagorean identity is a fundamental concept in trigonometry. It is given by the equation:
  • \(\cos^2 x + \sin^2 x = 1\)
This identity relates the squares of the sine and cosine of an angle, and it's a consequence of the Pythagorean theorem in a unit circle. When we apply this to our expression of \(\cos^4 x - \sin^4 x\), we see that the term \(\cos^2 x + \sin^2 x\) equals 1. Therefore, the expression becomes:
  • \(\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x\)
This simplification is crucial as it reduces the problem drastically by eliminating one of the operations. The Pythagorean identity is often used to simplify equations involving trigonometric functions.
Double Angle Identities
The double angle identities provide a way to express trigonometric functions of double angles in terms of single angles. A key double angle identity for cosine is:
  • \(\cos 2x = \cos^2 x - \sin^2 x\)
This identity is instrumental in converting expressions like \(\cos^2 x - \sin^2 x\) into \(\cos 2x\). In the context of our solution, once we simplify \(\cos^4 x - \sin^4 x\) to \(\cos^2 x - \sin^2 x\), we can replace \(\cos^2 x - \sin^2 x\) with \(\cos 2x\). This substitution simplifies the original expression to:
  • \(\cos^2 x - \sin^2 x - (\cos^2 x - \sin^2 x)\), which equals 0.
Using the double angle identity is a clever way to further reduce terms and see the final outcome clearly. Such identities are convenient for transforming and solving trigonometric expressions efficiently.

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