I know the solution to this question, which is very long.
Prove the trigonometric identity $$\cos2A+\cos2B-\cos2C=1-4\sin A\sin B\cos C \tag{$\star$}$$
where the angles are part of $\triangle ABC$
I would like to know:
Is it possible to prove $(\star)$ with the following trigonometric identities?
$$\sin2A+\sin2B+\sin2C=4\sin A\sin B\sin C$$
$$\cos2A+\cos2B+\cos2C=-1-4\cos A\cos B\cos C$$
My teacher told me such questions can be solved very quickly using these identities instead of the transformation formulas. However I just can figure out how to use them.
Best Answer
Using $$\cos(2x)=2\cos^2(x)-1$$ we have to prove $$\cos^2(A)+\cos^2(B)+\cos^2(C)=1-2\cos(A)\cos(B)\cos(C)$$ now use the theorem of cosines.