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Pythagorean Identities

Maths: Edexcel A Level Pure Maths, topic 5.4.2

The Pythagorean Identity

The definition of the unit circle gives rise to an important identity involving the trigonometric functions.

Unit circle

Unit circle

  • We defined the coordinates of the unit circle as being given by the functions:
  • x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta
  • Any point that lies on a circle is exactly one radius in distance away from the center of the circle.
  • This is equal to one for the unit circle.
Proof

Proof

  • By Pythagoras' theorem, we can write:
  • 1=x2+y2=(cos⁡θ)2+(sin⁡θ)21 = \sqrt{x^2+y^2} =\sqrt{(\cos\theta)^2 +( \sin\theta)^2}
  • Squaring both sides and using the notation (cos⁡θ)2=cos⁡2θ(\cos\theta)^2=\cos^2\theta gives the Pythagorean identity:
  • cos2θ+sin2θ=1cos^2\theta + sin^2\theta = 1
Tan and sec identity

Tan and sec identity

  • Dividing the Pythagorean identity through by cos⁡2θ\cos^2\theta we have:
  • 1+sin⁡2θcos⁡2θ=1cos⁡2θ1 + \frac{\sin^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}
  • We can then substitute the reciprocal identities as:
  • 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
Cot and cosec identity

Cot and cosec identity

  • Dividing the Pythagorean identity through by sin⁡2θ\sin^2\theta we have:
  • cos⁡2θsin⁡2θ+1=1sin⁡2θ\frac{\cos^2\theta}{\sin^2\theta} + 1= \frac{1}{\sin^2\theta}
  • We can then substitute the reciprocal identities as:
  • 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

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