Use the Pythagorean identity to show that the double angle formula for cosine can be written as
a) cos2x = 1 - 2sin^2x
b) cos2x = 2cos^2x - 1
sin^2x+ cos^2x=1
cos^2x+ sin^2x-2sin^2x=1-2sin^2x
cos^2x-sin^2x=1-2sin^2x
cos(2x)=1-2sin^2x
To use the Pythagorean identity to derive the double angle formula for cosine, we start with the Pythagorean identity itself:
cos^2x + sin^2x = 1
Now, let's rewrite cos^2x in terms of sin^2x:
cos^2x = 1 - sin^2x
Next, we will multiply both sides of the equation by 2:
2cos^2x = 2(1 - sin^2x)
Expanding the right side of the equation:
2cos^2x = 2 - 2sin^2x
Now, rearrange the equation:
2cos^2x - 1 = 1 - 2sin^2x
Therefore, the double angle formula for cosine can be written as:
a) cos2x = 1 - 2sin^2x
b) cos2x = 2cos^2x - 1
To use the Pythagorean identity to derive the double angle formula for cosine, we start with the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
Now, let's manipulate this equation to isolate sin^2(x):
sin^2(x) = 1 - cos^2(x)
Next, we want to express cos(2x) in terms of sin(x) and cos(x). The double angle formula for cosine states that:
cos(2x) = cos^2(x) - sin^2(x)
To substitute the expression for sin^2(x) from the Pythagorean identity into the double angle formula for cosine, we get:
cos(2x) = cos^2(x) - (1 - cos^2(x))
Simplifying this equation, we have:
cos(2x) = cos^2(x) - 1 + cos^2(x)
Now, combine like terms:
cos(2x) = 2cos^2(x) - 1
Therefore, the double angle formula for cosine can be written as:
cos(2x) = 2cos^2(x) - 1
So, option b) is correct.