Prove the identity
Cos²x - sin²x = 2cos²x - 1
To prove the identity, we will start with the left side of the equation and manipulate it until we obtain the right side.
Starting with the left side:
cos²x - sin²x
Using the Pythagorean identity:
cos²x - (1 - cos²x)
Expanding the expression:
cos²x - 1 + cos²x
Combine like terms:
2cos²x - 1
Therefore, the left side (cos²x - sin²x) can be simplified to the right side (2cos²x - 1).
To prove the identity:
Cos²x - sin²x = 2cos²x - 1
We can start by using the Pythagorean Identity for sine and cosine:
sin²x + cos²x = 1
Now, let's substitute this identity into the equation:
Cos²x - sin²x = 2cos²x - 1
(cos²x + sin²x) - sin²x = 2cos²x - 1
cos²x + sin²x - sin²x = 2cos²x - 1
cos²x = 2cos²x - 1
Next, let's simplify by moving all the terms to one side of the equation:
0 = 2cos²x - 1 - cos²x
Rearranging the terms:
0 = cos²x - 1
And finally, adding 1 to both sides of the equation:
1 = cos²x
Which is the same as:
cos²x = 1
Therefore, we have proven that Cos²x - sin²x = 2cos²x - 1.