Question

How do you prove `(4sintheta-costheta)(1-2sin^2 theta) = sin(4theta)`?

Answer 1

You don't

Explanation:

Using the double-angle formulas `sin(2x)=2sinxcosx` and `cos(2x)=1-2sin^2x`, let's work the right side of the equation.

`sin(4theta)=sin(2(2theta))=2sin(2theta)cos(2theta)=2(2sinthetacostheta)(1-2sin^2theta)=(4sinthetacostheta)(1-2sin^2theta)`

As far as I can tell, `(4sinthetacostheta)(1-2sin^2theta)!=(4sintheta-costheta)(1-2sin^2theta)` so the identity is false.