Factor the expression and use the fundamental identities to simplify.
1-2 cos^2x + cos^4x
this is just
(1-cos^2)^2 = (sin^2)^2 = sin^4
To factor and simplify the expression 1-2cos^2x+cos^4x using the fundamental identities, we can start by recognizing that cos^4x can be expressed as (cos^2x)^2.
So, the expression becomes 1-2cos^2x + (cos^2x)^2.
Now, let's manipulate the expression using the fundamental identities:
1 - 2cos^2x + (cos^2x)^2
= 1 - 2cos^2x + cos^2x * cos^2x
= 1 - 2cos^2x + (1 - sin^2x) * (1 - sin^2x)
= 1 - 2cos^2x + (1 - 2sin^2x + sin^4x)
Now, let's combine like terms:
= 1 - 2cos^2x + 1 - 2sin^2x + sin^4x
= 2 - 2cos^2x - 2sin^2x + sin^4x
Finally, we can simplify further by using the Pythagorean identity, sin^2x + cos^2x = 1:
= 2 - 2(1 - sin^2x) - 2sin^2x + sin^4x
= 2 - 2 + 2sin^2x - 2sin^2x + sin^4x
= -sin^4x + 2sin^2x
Therefore, the simplified expression is -sin^4x + 2sin^2x.