Question

Sin4x/1+cos4x*cos2x/1+cos2x=

Answer 1

Sin4x+cos4x*cos2x+cos2x
потому что деление на 1 ничего не изменяет

Answer 2

To simplify the expression, let's first focus on individual parts and then combine them:

1. Simplifying sin(4x) / (1 + cos(4x)): Using the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ), we can rewrite sin(4x) as 2sin(2x)cos(2x):

sin(4x) = 2sin(2x)cos(2x)

Now, the expression becomes:

2sin(2x)cos(2x) / (1 + cos(4x))

2. Simplifying cos(2x) / (1 + cos(2x)): To simplify this part, we can multiply the numerator and denominator by the conjugate of (1 + cos(2x)), which is (1 - cos(2x)):

cos(2x) * (1 - cos(2x)) / (1 + cos(2x)) * (1 - cos(2x))

Now, using the identity cos^2(θ) = 1 - sin^2(θ), we can simplify further:

cos(2x) * (1 - cos^2(2x)) / (1 - cos^2(2x))

Now, the expression becomes:

cos(2x) - cos^3(2x) / sin^2(2x)

3. Combining the two parts: Now that we have simplified both parts, let's combine them:

2sin(2x)cos(2x) / (1 + cos(4x)) * (cos(2x) - cos^3(2x) / sin^2(2x))

4. Further simplification: We can simplify this expression further by factoring out common terms:

2sin(2x)cos(2x) / (1 + cos(4x)) * cos(2x) * (1 - cos^2(2x)) / sin^2(2x)

Next, using the identity sin^2(θ) = 1 - cos^2(θ), we can simplify sin^2(2x) in the denominator:

2sin(2x)cos(2x) / (1 + cos(4x)) * cos(2x) * (1 - cos^2(2x)) / (1 - cos^2(2x))

Since (1 - cos^2(2x)) appears in both the numerator and denominator, they cancel out:

2sin(2x)cos(2x) / (1 + cos(4x)) * cos(2x)

Now, we can simplify further by canceling out a common factor, cos(2x), from the numerator and denominator:

2sin(2x) / (1 + cos(4x))

This is the simplified expression for the given expression sin(4x) / (1 + cos(4x)) * cos(2x) / (1 + cos(2x)).