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.