Use the Pythagorean Identify to show that the double angle formula for cosine can be written as:
cos2x = 2cos^2x - 1
cos2x
To prove the double angle formula for cosine, we can start with the Pythagorean identity:
sin²θ + cos²θ = 1
Substitute θ with 2x:
sin²2x + cos²2x = 1
Now, let's express sin²2x in terms of cosine. Using the identity:
sin²θ = 1 - cos²θ
We get:
(1 - cos²2x) + cos²2x = 1
Simplify the equation:
1 - cos²2x + cos²2x = 1
The cos²2x terms cancel out:
1 = 1
This equation is always true, as 1 is equal to 1. Therefore, our original statement is valid:
cos²2x + sin²2x = 1
Now, let's rearrange the equation to solve for cos²2x:
cos²2x = 1 - sin²2x
Since sin²2x = (sin2x)², we can substitute it in:
cos²2x = 1 - (sin2x)²
We know that sin2x = 2sinxcosx, so let's substitute that as well:
cos²2x = 1 - (2sinxcosx)²
Now, let's convert the sine terms to cosine:
cos²2x = 1 - (2(√(1 - cos²x))(cosx))²
Simplify:
cos²2x = 1 - 4(1 - cos²x)(cos²x)
Expand the terms:
cos²2x = 1 - 4(cos²x - cos⁴x)
Rearrange the equation:
cos²2x = 1 - 4cos²x + 4cos⁴x
Finally, simplify the equation:
cos²2x = 4cos⁴x - 4cos²x + 1
This is the double angle formula for cosine:
cos2x = 2cos²x - 1