How do you prove: Sin3θ = 3Sinθ−4Sin3θ?

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: Sin3θ = 3sinθ - 4sin³θ

Let us analyse this problem step by step

Explanation:

We can use the trigonometric formula,

sin (x+y) = sin x cos y + cos x sin y

x = 2θ and y = θ  [ 3θ can be written as sum of θ and 2θ]

Substituting the values of x and y we get,

sin(2θ+θ) = sin2θ cosθ + cos2θ sinθ 

⇒ sin3θ = (2sinθcosθ)cosθ + (cos²θ - sin²θ )sinθ  [Since, sin 2x = 2 sin x cos x and cos 2x = cos²x - sin²x]

⇒ sin 3θ = 2sinθcos²θ + cos²θsinθ - sin³θ

⇒ sin 3θ = 2sinθ (1-sin²θ) + ( 1-sin²θ) sinθ - sin³θ  [Since, cos²x = 1 - sin²x]

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

⇒ sin 3θ = 3sinθ - 4sin³θ

Hence, proved sin3θ = 3sinθ - 4sin³θ.