OK, I've been stuck on this question for like.. an hour. Prove how cos 2x = 1. Thanks for the help!
Nvm... lol I got it. There was a typo. Sorry!
No problem! It happens to everyone. If you ever come across a similar question in the future, here is how you can prove that cos(2x) equals 1:
To prove that cos(2x) equals 1, we can use the double-angle identity for cosine. The double-angle identity for cosine states that cos(2x) = cos²(x) - sin²(x).
If cos(2x) = 1, then we need to show that cos²(x) - sin²(x) = 1.
To do this, we'll use another trigonometric identity called the Pythagorean identity, which states that sin²(x) + cos²(x) = 1.
Rearranging the Pythagorean identity, we have cos²(x) = 1 - sin²(x).
Now let's substitute this value of cos²(x) into the double-angle identity for cosine:
cos(2x) = cos²(x) - sin²(x)
cos(2x) = 1 - sin²(x) - sin²(x)
cos(2x) = 1 - 2sin²(x)
Now, if cos(2x) = 1, then we need to show that 1 - 2sin²(x) = 1.
Simplifying this equation, we have:
1 - 2sin²(x) = 1
-2sin²(x) = 0
sin²(x) = 0
Taking the square root of both sides, we get:
sin(x) = 0
This means that x can be any integer multiple of π because sin(0) = 0 at those points.
Hence, cos(2x) does not equal 1 for all values of x; it equals 1 only when sin(x) equals 0.