Verify the identities.
Cos^2x - sin^2x = 2cos^2x - 1
When verifying identities, can I work on both side?
Ex.
1 - sin^2x - sin^2x = 1 - 2sin^2x
1 - 2sin^2x = 1 - 2sin^2x
cos ^ 2 ( x ) - sin ^ 2 ( x ) = 2 cos ^ 2 ( x ) - 1 Subtract cos ^ 2 ( x ) to both sides
cos ^ 2 ( x ) - sin ^ 2 ( x ) - cos ^ 2 ( x ) = 2 cos ^ 2 ( x ) - 1 - cos ^ 2 ( x )
- sin ^ 2 ( x ) = cos ^ 2 ( x ) - 1 Add 1 to both sides
- sin ^ 2 ( x ) + 1 = cos ^ 2 ( x ) - 1 + 1
1 - sin ^ 2 ( x ) = cos ^ 2 ( x )
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Remark:
1 - sin ^ 2 ( x ) = cos ^ 2 ( x )
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So :
cos ^ 2 ( x ) = cos ^ 2 ( x )
Yes, when verifying identities, you can work on both sides of the equation. The goal is to manipulate the expressions on one side of the equation until it is equivalent to the other side.
Let's start by working on the left side of the equation:
cos^2x - sin^2x
We can rewrite cos^2x as (1 - sin^2x) using the Pythagorean identity: sin^2x + cos^2x = 1.
So, the left side becomes:
(1 - sin^2x) - sin^2x
Simplifying further:
1 - 2sin^2x
Now, let's work on the right side of the equation:
2cos^2x - 1
We can rewrite cos^2x as 1 - sin^2x (using the same Pythagorean identity).
So, the right side becomes:
2(1 - sin^2x) - 1
Simplifying further:
2 - 2sin^2x - 1
1 - 2sin^2x
As you can see, after simplifying both sides, we get the same expression: 1 - 2sin^2x.
Therefore, we have verified that cos^2x - sin^2x is equal to 2cos^2x - 1.