Structure Recognition Math Example 2

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Example 2

medium
Identify the structure in sin⁡2θ+cos⁡2θ\sin^2\theta + \cos^2\theta and use it to simplify sin⁡4θ−cos⁡4θsin⁡2θ−cos⁡2θ\dfrac{\sin^4\theta - \cos^4\theta}{\sin^2\theta - \cos^2\theta}.

Solution

  1. 1
    Recognise the numerator as a difference of squares: a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b) with a=sin⁡2θa = \sin^2\theta, b=cos⁡2θb = \cos^2\theta.
  2. 2
    So sin⁡4θ−cos⁡4θ=(sin⁡2θ−cos⁡2θ)(sin⁡2θ+cos⁡2θ)\sin^4\theta - \cos^4\theta = (\sin^2\theta - \cos^2\theta)(\sin^2\theta + \cos^2\theta).
  3. 3
    Divide by sin⁡2θ−cos⁡2θ\sin^2\theta - \cos^2\theta (assuming sin⁡2θ≠cos⁡2θ\sin^2\theta \ne \cos^2\theta): result =sin⁡2θ+cos⁡2θ=1= \sin^2\theta + \cos^2\theta = 1.

Answer

sin⁡4θ−cos⁡4θsin⁡2θ−cos⁡2θ=1\frac{\sin^4\theta-\cos^4\theta}{\sin^2\theta-\cos^2\theta} = 1
Recognising the difference-of-squares structure converts a complicated expression into a known identity. The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1 then completes the simplification.

About Structure Recognition

The skill of identifying that a given mathematical expression or problem belongs to a known family or matches a recognizable pattern.

Learn more about Structure Recognition →

More Structure Recognition Examples

Example 1 easy

Recognise and name the algebraic structure in [formula], then factorise.

Example 3 easy

Identify the structure and factorise: [formula].

Example 4 medium

Recognise the structure in the sum [formula] and compute it using the geometric series formula.

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