Proving the Double Angle Identity for Cosine: A Trigonometry Tutorial

cos(2x) = 2cos^2 – 1

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The given equation cos(2x) = 2cos^2 (x) – 1 is a double angle identity in trigonometry.

To prove this identity, we start with the double-angle formula for cosine, which states that:

cos(2x) = cos(x + x)

Using the sum formula for cosine, we can expand this expression as:

cos(2x) = cos(x)cos(x) – sin(x)sin(x)

Now, we know that cos^2 (x) + sin^2 (x) = 1, so we can substitute cos^2 (x) = 1 – sin^2 (x) into the equation:

cos(2x) = (1 – sin^2 (x)) – sin^2 (x)

Simplifying the right-hand side:

cos(2x) = 1 – 2 sin^2 (x)

Now, we have to use the double angle identity for cosine, which is:

cos(2x) = 2cos^2 (x) – 1

Therefore, we can equate the two expressions we have derived for cos(2x) and simplify to get:

1 – 2 sin^2 (x) = 2cos^2 (x) – 1

Rearranging and solving for cos^2 (x):

2cos^2 (x) = 1 + cos(2x)

cos^2 (x) = (1 + cos(2x))/2

This proves the double angle identity cos(2x) = 2cos^2 (x) – 1.

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