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Factor the following expressions and use the fundamental identities to simplify. a) cosx−2/cos ^2x−4 b) sin^4 x−cos^4x

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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).

User Sebbo
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