Use the double angle formulas to verify the identity. `sin4x = 8cos^3x*sinx - 4cosx*sinx`

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Start with the left hand side and try to get to the right hand side:

sin(4x)=sin[2(2x)]                    Use sin2A=2sinAcosA:

=2sin(2x)cos(2x)                      Again use sin2A=2sinAcosA:

=2[2sin(x)cos(x)cos(2x)]

=4sin(x)cos(x)cos(2x)

Here we have a choice for cos(2x) -- choose 2cos^2x-1

=4sin(x)cos(x)[2cos^2(x)-1]     Multiply to get:

=8cos^3(x)sin(x)-4sin(x)cos(x) as required.

Sources:

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