Double Angle Trigonometric Identities

The Double Angle Identities:

sin(2θ)= sin(θ+θ)= sinθcosθ + cosθsinθ= 2sinθcosθ

cos(2θ)= cos(θ+θ)= cosθcosθ-sinθsinθ= cos ²θ-sin ²θ

cos(2θ)= cos ²θ-sin ²θ= 1-sin ²θ 2 cos ²θ

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

**NOTE**

csc²θ= 1/sin²θ

sec²θ=1/cos²θ

cot²θ=1/tan²θ

Example: verify

sin3 x = -4sin³ x+3 sin x

LHS= sin 3x
=sin(2x+ x)
=sin 2x cos x + cos 2x sin x
=(2sin x cos x)cos x+ ( cos ² x-sin ² x)sin x
=2 sin x cos ² x + cos ² x sin x - sin 3x
=2sin x(1-sin ² x)+(1-sin ² x)sin x - sin 3x
=2sin x- 2 sin³ x+ sin x -sin³ x- sin 3x
LHS=-4sin³ x + 3 sin x = RHS