What are the formulae of 1 + cos(2x) and 2(1 + cos(2x))?

The formula for 1 + cos(2x) can be derived from the trigonometric identity for cosine. The expression simplifies using the double angle formula of cosine, which states:

• cos(2x) = 2cos²(x) – 1

Using this identity, we rewrite:

• 1 + cos(2x) = 1 + (2cos²(x) – 1) = 2cos²(x)

So, 1 + cos(2x) = 2cos²(x).

Next, let’s consider the expression 2(1 + cos(2x)). From our earlier result, we can substitute:

• 2(1 + cos(2x)) = 2(2cos²(x)) = 4cos²(x)

In summary:

• The formula for 1 + cos(2x) is 2cos²(x).
• The formula for 2(1 + cos(2x)) is 4cos²(x).