Answer:
Step-by-step explanation:
In double angle, sin2x = sin(x+x) = sinxcosx+cosxsinx
sin2x = 2sinxcosx ... 1
Applying this formula to prove that sin(4x) = 4 sinx cosx(1 − 2sin2x is shown below;
sin(4x) = sin(2x+2x)
= sin2xcos2x+cox2xsin2x
sin4x = 2sin2xcos2x ..2
also cos2x = cos(x+x) = cosxcox-sinxsinx
cos 2x = cos²x - sin²x ...3
Substituting equation 1 and 3 into 2, we will have;
sin4x = 2(2sinxcosx(cos²x - sin²x ))
sin4x = 4sinxcosx(cos²x - sin²x )
From sin²x+cos²x =1; cos²x = 1-sin²x
Substituting the expression into the resulting equation will give;
sin4x = 4sinxcosx(1-sin²x - sin²x )
sin4x = 4sinxcosx(1-2sin²x) Verified!
