cotxxot2x -cot2xcot3x - cot3xcotx =??
plese help ME
I don't see any shortcuts here, it appears to bje a plug and grind it out use of the angle formulas.
To simplify the given expression cot(x) * cot(2x) - cot(2x) * cot(3x) - cot(3x) * cot(x), we can use trigonometric identities.
1. Start by simplifying each term using the reciprocal identity for cotangent:
cot(x) = 1/tan(x)
cot(2x) = 1/tan(2x)
cot(3x) = 1/tan(3x)
Now the expression becomes:
(1/tan(x)) * (1/tan(2x)) - (1/tan(2x)) * (1/tan(3x)) - (1/tan(3x)) * (1/tan(x))
2. Next, use the identity for tangent:
tan(a) = sin(a)/cos(a)
Applying this identity to each term:
(1/(sin(x)/cos(x))) * (1/(sin(2x)/cos(2x))) - (1/(sin(2x)/cos(2x))) * (1/(sin(3x)/cos(3x))) - (1/(sin(3x)/cos(3x))) * (1/(sin(x)/cos(x)))
3. Simplify further:
Flip the division in each term and multiply:
(cos(x)/sin(x)) * (cos(2x)/sin(2x)) - (cos(2x)/sin(2x)) * (cos(3x)/sin(3x)) - (cos(3x)/sin(3x)) * (cos(x)/sin(x))
4. Cancel out the common factors:
(cos(x) * cos(2x)) - (cos(2x) * cos(3x)) - (cos(3x) * cos(x))
Rewrite the terms by rearranging the cosine factors:
cos(x) * [cos(2x) - cos(3x)] - cos(3x) * cos(x)
5. Simplify the remaining terms:
Use the cosine subtraction identity:
cos(p) - cos(q) = -2 * sin((p+q)/2) * sin((p-q)/2)
Applying this identity:
cos(2x) - cos(3x) = -2 * sin((2x+3x)/2) * sin((2x-3x)/2)
cos(2x) - cos(3x) = -2 * sin(5x/2) * sin(-x/2)
cos(2x) - cos(3x) = -2 * sin(5x/2) * (-sin(x/2))
6. Substitute back into the simplified expression:
cos(x) * [-2 * sin(5x/2) * (-sin(x/2))] - cos(3x) * cos(x)
cos(x) * 2 * sin(5x/2) * sin(x/2) - cos(3x) * cos(x)
So, the simplified expression is cos(x) * 2 * sin(5x/2) * sin(x/2) - cos(3x) * cos(x).