Does $-2 \int (\cos^2\,x-\sin^2\,x)\,\sin\,2x\,dx$ have multiple answers?

Question

After working $-2 \displaystyle\int (\cos^2\,x-\sin^2\,x)\,\sin\,2x\,dx$, I got $\frac{-\sin^2(2x)}{2}+c$. The answer I was given was $\frac{\cos^2(2x)}{2}+c$. Are these both legitimate solutions, depending on if sine or cosine is chosen for substitution?

Answer 1

$E=\frac{-\sin^2(2x)}{2}+c=\frac{-1+\cos^2(2x)}{2}+c$ $E=\frac{\cos^2(2x)}{2}+c-\frac 1 2=\frac{\cos^2(2x)}{2}+k$

Where $ k=c-\frac 1 2$

These are the same answers.