Question 32478
Cos2A = Cos(A+A)

we know the formula for Cos(A+B)=CosACosB-SinASinB
therefore Cos(A+A)= CosACosA - SinASinA
                  = Cos^2A - Sin^2A
WE also know that Cos^2A + Sin^2A = 1   
          therfore Sin^2A = 1 - Cos^2A

Replacing the value of Sin^2A 

Cos(A+A)= Cos^2A - (1 - Cos^2A)
Cos2A = Cos^2A - 1 +Cos^2A

Hence Cos2A = 2Cos^2A - 1

Thus Proved.