Final answer:
The statement 1 + Cos 2A = 2(Cos^2 A + Sin^2 A) is proven true by substituting Cos 2A with 2 Cos^2A - 1 and then using the Pythagorean identity to replace 1 with Cos^2A + Sin^2A on the left side of the equation.
Step-by-step explanation:
To prove the identity 1 + Cos 2A = 2(Cos2 A + Sin2 A), we can make use of trigonometric identities that are already known. The identity Cos 2A can be written in three different ways, usually as Cos2A - Sin2A, 2 Cos2A - 1, or 1 - 2 Sin2A. The most convenient form for our proof is the second one, 2 Cos2A - 1, because we have the term 2 Cos2A present which is related to the term we want to prove.
First, we substitute Cos 2A with 2 Cos2A - 1 in our original equation:
1 + Cos 2A = 1 + (2 Cos2A - 1)
Now, when we simplify the right side of the equation, the 1 and -1 cancel out:
1 + Cos 2A = 2 Cos2A
Recognizing that Cos2A + Sin2A equals 1 according to the Pythagorean identity, we replace the 1 on the left side with Cos2A + Sin2A:
1 + Cos 2A = 2(Cos2A) + 2(Sin2A)
which simplifies to:
1 + Cos 2A = 2(Cos2A + Sin2A)
It's now been proven that the original statement is true using trigonometric identities.