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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