Math Problem Statement

Prove the identity: cos(4θ) + 4cos(2θ) ≡ 8cos^4(θ) - 3.

Solution

We are tasked with proving the identity:

cos(4θ)+4cos(2θ)8cos4(θ)3.\cos(4\theta) + 4\cos(2\theta) \equiv 8\cos^4(\theta) - 3.

Step 1: Express cos(4θ)\cos(4\theta) and cos(2θ)\cos(2\theta) in terms of cos(θ)\cos(\theta).

  • We use the known double-angle identities: cos(2θ)=2cos2(θ)1,\cos(2\theta) = 2\cos^2(\theta) - 1, cos(4θ)=2cos2(2θ)1=2(2cos2(θ)1)21.\cos(4\theta) = 2\cos^2(2\theta) - 1 = 2(2\cos^2(\theta) - 1)^2 - 1. Let's simplify cos(4θ)\cos(4\theta): cos(4θ)=2(4cos4(θ)4cos2(θ)+1)1.\cos(4\theta) = 2(4\cos^4(\theta) - 4\cos^2(\theta) + 1) - 1. cos(4θ)=8cos4(θ)8cos2(θ)+21=8cos4(θ)8cos2(θ)+1.\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 2 - 1 = 8\cos^4(\theta) - 8\cos^2(\theta) + 1.

Step 2: Express the left-hand side of the identity.

We now have: cos(4θ)+4cos(2θ)=(8cos4(θ)8cos2(θ)+1)+4(2cos2(θ)1).\cos(4\theta) + 4\cos(2\theta) = (8\cos^4(\theta) - 8\cos^2(\theta) + 1) + 4(2\cos^2(\theta) - 1). Simplifying the terms: cos(4θ)+4cos(2θ)=8cos4(θ)8cos2(θ)+1+8cos2(θ)4.\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1 + 8\cos^2(\theta) - 4. cos(4θ)+4cos(2θ)=8cos4(θ)3.\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 3.

Step 3: Conclusion

The identity is verified as:

cos(4θ)+4cos(2θ)=8cos4(θ)3.\cos(4\theta) + 4\cos(2\theta) = 8\cos^4(\theta) - 3.

Would you like further details on any part of the proof? Here are some related questions for further exploration:

  1. What are other forms of the double-angle identities?
  2. How can the identity for sin(4θ)\sin(4\theta) be derived similarly?
  3. What is the general form of the multiple-angle identities?
  4. How would you approach proving an identity involving tan(θ)\tan(\theta)?
  5. Can trigonometric identities be useful in solving integrals?

Tip: Always check if a trigonometric identity can be simplified using standard known formulas such as double-angle, half-angle, or sum-to-product identities.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities

Formulas

cos(2θ) = 2cos^2(θ) - 1
cos(4θ) = 2cos^2(2θ) - 1

Theorems

Double Angle Identity
Trigonometric Simplification

Suitable Grade Level

Grade 10-12 or College level