Help me prove this identity?

Question

sin(3x)=-4sin^3(x)+3sin(x)

Answer 1

sin(A+B) = sinAcosB + cosAsinB
sin^2 A + cos^2 A = 1

cos(A+B) = cosAcosB - sinAsinB

sin(2x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x)

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

Answer 2

Consider the Left Hand side:
sin(3x) = sin(2x + x).. opening this up will give you:
sin(2x)cosx + cos(2x)sinx..... again sin(2x) = sin(x + x) = 2sinxcosx and cos(2x) = cos(x + x)= cos^2(x) - sin^2(x)
so we get:
(2sinxcosx)cosx + {cos^2(x) - sin^2(x)}sinx
2sinxcos^2(x) + sinxcos^2(x) - sin^3(x)
3sinxcos^2(x) - sin^3(x)... now...... cos^2(x) = 1 -sin^2(x)... so
3sinx{1-sin^2(x)} - sin^3(x)
3sinx - 3sin^3(x) - sin^3(x)
3sinx - 4 sin^3(x) or
-4sin^3(x) + 3sin(x).... Hence proved

Answer 3

sin(3x) = 4sin^3(x) + 3sin(x)

Choosing the left hand side,

LHS = sin(3x)

Decompose 3x into 2x + x

LHS = sin(2x + x)

Use the sine addition identity, sin(a + b) = sin(a)cos(b) + sin(b)cos(a)

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

Use the double angle identities, sin(2x) = 2sin(x)cos(x), and cos(2x) = cos^2(x) - sin^2(x).

LHS = 2sin(x)cos(x)cos(x) + sin(x) ( cos^2(x) - sin^2(x) )

Expanding,

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

Use the identity cos^2(x) = 1 - sin^2(x).

LHS = 3sin(x) [ 1 - sin^2(x) ] - sin^3(x)

Expand.

LHS = 3sin(x) - 3sin^3(x) - sin^3(x)

Group like terms.

LHS = 3sin(x) - 4sin^3(x)

Swap the terms.

LHS = -4sin^3(x) + 3sin(x) = RHS

Answer 4

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

LHS = RHS

Answer 5

sin(3x)=sin(2x+x)
=sin(2x)cosx+cos(2x)sinx
=2sinxcos^2(x)+(1-2sin^2(x))sinx
=2sinx(1-sin^2(x))+sinx-2sin^3(x)
=-4sin^3(x)+3sin(x)