Double Angle Formulas of Sin

The sum formula of sine function is,

sin (A + B) = sin A cos B + cos A sin B

When A = B, the above formula becomes,

sin (A + A) = sin A cos A + cos A sin A

sin 2A = 2 sin A cos A

Let us derive an alternate formula for sin 2A in terms of tan using the Pythagorean identity sec²A = 1 + tan²A.

sin2A=2sinAcosA=2sinAcos2AcosA=2sinAcosA⋅cos2A=2tanA⋅1sec2A=2tanA1+tan2Asin⁡2A=2sin⁡Acos⁡A=2sin⁡Acos2⁡Acos⁡A=2sin⁡Acos⁡A⋅cos2⁡A=2tan⁡A⋅1sec2⁡A=2tan⁡A1+tan2⁡A

Thus, the double angle formulas of sine function are

sin 2A = 2 sin A cos A (or) (2 tan A) / (1 + tan²A)