1. By considering

   $$\sin (A+B)$$

   when $A=x$ and $B=x,$ show that

   $$\sin (2x)=2\sin x\cos x$$

2. By considering

   $$\cos (A+B)$$

   when $A=x$ and $B=x,$ show that

   $$\cos (2x)={\cos }^{2}x-{\sin }^{2}x$$

3. Prove, additionally, that

   $$\cos (2x)=2{\cos }^{2}x-1$$

   and

   $$\cos (2x)=1-2{\sin }^{2}x$$

For (a), start with

Let $A=x$ and $B=x$.

For (b), start with

For (c), recall that in (b), we showed that

But we also know that

and this can be used to eliminate either ${\sin }^2$ or ${\cos }^2$ from the double angle formula for $\cos .$