How is $ 4 \cos^2 (t/2) \sin(1000t) = 2 \sin(1000t) + 2\sin(1000t)\cos t$?

Question

How is $\; 4 \cos^2 (t/2) \sin(1000t) = 2 \sin(1000t) + 2\sin(1000t)\cos t\,$? This is actually part of a much bigger physics problem, so I need to solve it from the LHS quickly. Is there an easy method by which I can do this?

Answer 1

Use the linearisation formula: $\qquad\cos^2x=\dfrac{1+\cos 2x}2$.

Answer 2

Just use the fact that$$\cos(t)=\cos^2\left(\frac t2\right)-\sin^2\left(\frac t2\right)=2\cos^2\left(\frac t2\right)-1.$$