simplify (cos4x-cos2x)/(sin4x+sin2x) the answer choices are : cotx, tanx, cotxtanx, or 1

let x be 45 deg

(cos180-cos90)/(sin180+sin90)=-1/(1)=-1

so none of the answers are correct, as ctn(45), tan(45), ctn(45)*tan(45), or 1 are not equal to -1

another angle: let x be 30 deg
(cos120-cos60)/(sin120+sin(60)=(-.5-.5)/(.866+.866)=-.577
lets check given answers:
ctn30=1.73
tan30=.577
ctn30*tan30=1
1
none of these are equal to -.577

To simplify the expression (cos4x - cos2x) / (sin4x + sin2x), we can make use of the trigonometric identities.

Firstly, let's simplify the numerator (cos4x - cos2x):

Using the double-angle formula, we can write cos2x as 2cos²x - 1. So, we have:

cos4x - cos2x = cos4x - (2cos²x - 1)

Next, let's simplify the denominator (sin4x + sin2x):

Using the double-angle formula, we can write sin2x as 2sinx*cosx. So, we have:

sin4x + sin2x = sin4x + 2sinx*cosx

Now, we substitute these simplifications back into the original expression:

[(cos4x - (2cos²x - 1)) / (sin4x + 2sinx*cosx)]

Now, let's further simplify:

[(cos4x - 2cos²x + 1) / (sin4x + 2sinx*cosx)]

We can factor out a common factor of (cosx) in the numerator and (sinx) in the denominator:

[cosx * (cos3x - 2cosx + 1) / (sinx * (sin3x + 2cos²x))]

Now, let's simplify the remaining terms:

cos3x - 2cosx + 1 can be simplified using the sum-to-product formula, which states that:

cosA - cosB = -2sin[(A+B)/2] * sin[(A-B)/2].

Applying this formula to our expression, we have:
cos3x - 2cosx + 1 = -2sin(4x/2) * sin(2x/2)

Simplifying this further, we have:
cos3x - 2cosx + 1 = -2sin(2x) * sin(x)

After these simplifications, our expression becomes:

[cosx * (-2sin(2x) * sin(x)) / (sinx * (sin3x + 2cos²x))]

Now, we notice that sinx cancels out in the numerator and denominator:

[-2cosx * sin(2x)] / (sin3x + 2cos²x)

Finally, using the identity tanx = sinx / cosx, we can rewrite the expression as:

[-2cosx * sin(2x)] / [sin3x + 2(1 - sin²x)]

[-2cosx * sin(2x)] / [sin3x + 2 - 2sin²x]

[-2cosx * sin(2x)] / [sin3x - 2sin²x + 2]

This gives us the simplified expression for (cos4x - cos2x) / (sin4x + sin2x). However, none of the provided answer choices (cotx, tanx, cotxtanx, or 1) match this simplified expression.