sin3x formula proof | Sin 3x = 3Sin x - 4Sin³x trigonometric identity Proof

Sin 3x = 3Sin x -  4Sin³x  trigonometric identity Proof 

To prove the formula for sin(3x) step-by-step, we can use the angle addition identities and the double angle identity for sine. Here's the proof:

Solution :-

Start with the angle addition identity for sine:

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

Let A = 2x and B = x. Then the equation becomes:

sin(2x + x) = sin(2x)cos(x) + cos(2x)sin(x).

Expand cos(2x) using the double angle identity for cosine:

cos(2x) = cos^2(x) - sin^2(x).

Substitute the expanded expression into the equation:

sin(3x) = sin(2x)cos(x) + (cos^2(x) - sin^2(x))sin(x).

Simplify the expression by substituting sin(2x) = 2sin(x)cos(x):

sin(3x) = 2sin(x)cos(x)cos(x) + (cos^2(x) - sin^2(x))sin(x).

Rearrange the terms:

sin(3x) = 2sin(x)cos^2(x) + sin(x)cos^2(x) - sin^3(x).

Combine like terms:

sin(3x) = sin(x)cos^2(x) + 2sin(x)cos^2(x) - sin^3(x).

Simplify further:

sin(3x) = 3sin(x)cos^2(x) - sin^3(x).

And that's the proof for the formula of sin(3x) using trigonometric identities and the angle addition formula for sine.

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