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Trigonometric Identities and Equations, Page 19

Derivation of Double-Angle Identities

Let’s look at the Double-Angle Identities first to see how they are derived.

To find sin2θ = 2sinθ cosθ:

sin2θ = sin(θ + θ); sin2θ = sinθ cosθ + cosθ sinθ; sin2θ = 2sinθ cosθ

To find cos2θ = (cosθ)^2 - (sinθ)^2:

cos2θ = cos(θ + θ); cos2θ = cosθ cosθ - sinθ sinθ; cos2θ = (cosθ)^2 - (sinθ)^2

To find cos2θ = 2(cosθ)^2 - 1 :

Take cos2θ = (cosθ)^2 - (sinθ)^2 and substitute in (sinθ)^2 = 1 - (cosθ)^2 . 

cos2θ = (cosθ)^2 - (1 - (cosθ)^2); cos2θ = (cosθ)^2 - 1 + (cosθ)^2; cos2θ = 2(cosθ)^2 - 1

To find cos2θ = 1 - 2(sinθ)^2:

Take cos2θ = (cosθ)^2 - (sinθ)^2 and substitute in (cosθ)^2 = 1 - (sinθ)^2. 

cos2θ = (1- (sinθ)^2) - (sinθ)^2; cos2θ = 1 - (sinθ)^2

To find tan2θ = 2tanθ/(1 - (tanθ)^2):

tan2θ = tan(θ + θ); tan2θ = (tanθ + tanθ)/(1 - tanθ tanθ ); tan2θ = 2tanθ/(1 - (tanθ)^2)