Answer:
a) To factor and simplify the expression (cos(x) - 2) / (cos^2(x) - 4), we'll use the fundamental identity cos^2(x) - sin^2(x) = 1.
First, let's factor the numerator:
cos(x) - 2 = -(2 - cos(x))
Next, let's factor the denominator using the difference of squares:
cos^2(x) - 4 = (cos(x) + 2)(cos(x) - 2)
Now we can simplify the expression:
(cos(x) - 2) / (cos^2(x) - 4) = -(2 - cos(x)) / ((cos(x) + 2)(cos(x) - 2))
Notice that (cos(x) - 2) in the numerator and (cos(x) - 2) in the denominator can cancel out:
-(2 - cos(x)) / ((cos(x) + 2)(cos(x) - 2)) = -1 / (cos(x) + 2)
Therefore, the simplified expression is -1 / (cos(x) + 2).
b) To factor and simplify the expression sin^4(x) - cos^4(x), we'll use the difference of squares identity sin^2(x) - cos^2(x) = sin(x) + cos(x) and the Pythagorean identity sin^2(x) + cos^2(x) = 1.
First, let's rewrite the expression using the difference of squares identity:
sin^4(x) - cos^4(x) = (sin^2(x) - cos^2(x))(sin^2(x) + cos^2(x))
Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can simplify further:
(sin^2(x) - cos^2(x))(sin^2(x) + cos^2(x)) = (sin(x) + cos(x))(1)
The expression simplifies to sin(x) + cos(x).
Therefore, the simplified expression is sin(x) + cos(x).