How do you prove sin 3x = 3sin x - 4sin³ x?

Trigonometric functions (also called circular functions or angle functions)  are real functions that relate an angle of a right-angled triangle to the ratio of the other two side lengths.

Answer: sin 3x = 3 sinx - 4 sin³x

Let us analyse this problem step by step

Explanation:

To find the formula for sin 3θ, we can use sin (x + y) = sin x cos y + cos x sin y, where x is 2θ and y is θ.     [ 3θ can be written as sum of θ and 2θ]

sin (2θ + θ) = sin 2θ cos θ + cos 2θ sin θ

sin 3θ = (2 sin θ cos θ) cos θ + (cos²θ - sin²θ) sin θ

sin 3θ = 2 sin θ cos²θ + cos²θ sinθ - sin³θ

sin 3θ = 2 sin θ (1 - sin²θ) + (1 - sin²θ) sinθ - sin³θ

sin 3θ = 2 sin θ - 2 sin³θ + sin θ - sin³θ - sin³θ

sin 3θ = 3 sin θ - 4 sin³θ

Thus, sin 3x = 3 sinx - 4 sin³x