True or False cos^2x=(1-cos2x)/2
The Correct Answer and Explanation is :
The given equation, ( \cos^2(x) = \frac{1 – \cos(2x)}{2} ), is True.
Explanation:
This equation is a trigonometric identity derived from the double angle formula for cosine. The double angle formula for cosine states:
[
\cos(2x) = 2\cos^2(x) – 1
]
We can rearrange this equation to express ( \cos^2(x) ) in terms of ( \cos(2x) ):
[
\cos(2x) = 2\cos^2(x) – 1
]
Now, solve for ( \cos^2(x) ):
[
2\cos^2(x) = \cos(2x) + 1
]
[
\cos^2(x) = \frac{\cos(2x) + 1}{2}
]
We can also rewrite this as:
[
\cos^2(x) = \frac{1 – \cos(2x)}{2}
]
Hence, the given equation is verified.
Why this works:
This identity is useful in trigonometric integrals, calculus, and simplifying expressions where powers of trigonometric functions are involved. The equation is particularly helpful in transforming functions of squared trigonometric functions into functions with double angles, making them easier to integrate or differentiate.
In a geometric sense, the identity arises because the cosine function is periodic and symmetric, and the relationship between ( \cos(x) ) and ( \cos(2x) ) is based on how the cosine of an angle compares to the cosine of twice that angle. The factor of ( \frac{1}{2} ) helps to normalize the relationship between these angles, balancing the double angle’s effect.
This identity is part of the Pythagorean identities and double angle identities, both essential in trigonometry for simplifying and transforming expressions.