cos^4 x - sin^4 x = 1 - 2sin^2 x
Prove
Work with only one side to get it to match the other. It's critical you alter only ONE side.
What progress have you made, and where are you having problems?
I am completely stumped. I've never worked with ^4 before and I don't know how to factor it out.
Break cos^4(x) into cos^2(x)*cos^2(x).
To prove the equation cos^4(x) - sin^4(x) = 1 - 2sin^2(x), we will start by expressing both sides of the equation as a difference of squares.
First, let's write cos^4(x) as (cos^2(x))^2 and sin^4(x) as (sin^2(x))^2. This allows us to rewrite the equation as:
(cos^2(x))^2 - (sin^2(x))^2 = 1 - 2(sin^2(x))
Next, we can use the difference of squares identity, which states that a^2 - b^2 = (a + b)(a - b). Applying this identity to the left-hand side of the equation, we have:
[(cos^2(x)) + (sin^2(x))][(cos^2(x)) - (sin^2(x))] = 1 - 2(sin^2(x))
Since (cos^2(x)) + (sin^2(x)) is 1 (by the Pythagorean Identity), we can simplify the equation to:
(1)[(cos^2(x)) - (sin^2(x))] = 1 - 2(sin^2(x))
Next, we simplify the left-hand side to:
(cos^2(x)) - (sin^2(x)) = 1 - 2(sin^2(x))
Now, using the Pythagorean Identity again, we know that (cos^2(x)) - (sin^2(x)) is equal to cos(2x), so we can substitute this into the equation:
cos(2x) = 1 - 2(sin^2(x))
Finally, by manipulating the right-hand side of the equation, we can express sin^2(x) in terms of cos(2x):
cos(2x) = 1 - 2(1 - cos^2(x))
Simplifying further:
cos(2x) = 1 - 2 + 2cos^2(x)
cos(2x) = -1 + 2cos^2(x)
Now, using the double-angle formula for cosine, which states that cos(2x) = 2cos^2(x) - 1, we can see that both sides of the equation are equal:
2cos^2(x) - 1 = -1 + 2cos^2(x)
This proves that cos^4(x) - sin^4(x) = 1 - 2sin^2(x).