Double-Angle Identities Math Example 3

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Example 3

medium
Simplify cos⁡2(x)−sin⁡2(x)\cos^2(x) - \sin^2(x) using a double-angle identity.

Solution

  1. 1
    Recognize this as the double-angle formula for cosine: cos⁡(2x)=cos⁡2(x)−sin⁡2(x)\cos(2x) = \cos^2(x) - \sin^2(x).
  2. 2
    Therefore cos⁡2(x)−sin⁡2(x)=cos⁡(2x)\cos^2(x) - \sin^2(x) = \cos(2x).

Answer

cos⁡(2x)\cos(2x)
The expression cos⁡2x−sin⁡2x\cos^2 x - \sin^2 x is exactly the double-angle formula for cosine. Recognizing these patterns allows you to simplify expressions and solve equations more efficiently.

About Double-Angle Identities

Formulas expressing sin⁡(2θ)\sin(2\theta), cos⁡(2θ)\cos(2\theta), and tan⁡(2θ)\tan(2\theta) in terms of single-angle trig functions.

Learn more about Double-Angle Identities →

More Double-Angle Identities Examples

Example 1 easy

If [formula], find [formula] using the double-angle formula.

Example 2 medium

Find [formula] given that [formula] and [formula] is in Quadrant I.

Example 4 hard

Solve [formula] for [formula].

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