cotxcot2x-cot2xcot3x-cot3xcotx=1
are we solving ?
cotxcot2x-cot2xcot3x-cot3xcotx=1
(cosx/sinx)(cos2x/sin2x) - (cos2x/sin2x)(cos3x/sin3x) - (cos3x/sin3x)(cosx/sinx) = 1
common denominator is sinx(sin2x)(sin3x), so
[ cosxcos2xsin3x - sinxcos2xcos3x - sin2xcos3xcosx ] / (sinxsin2xsin3x) = 1
cross-multiply, then bring sinxsin2xsin3x back to left side
cosxcos2xsin3x - sinxcos2xcos3x - sin2xcos3xcosx - sinxsin2xsin3x = 0
factor sin3x from 1st and 4th, cos3 from 2nd and 3rd
sin3x(cosxcos2x - sinxsin2x) - cos3x(sinxcos2x + cosxsin2x) = 0
sin3x(cos(x+2x)) - cos3x(sin(x+2x)) = 0
sin3xcos3x - cos3xsin3x = 0
0 = 0
Since this is a true statement, the original equation must have been an identity.
To solve the given equation cot(x) * cot(2x) - cot(2x) * cot(3x) - cot(3x) * cot(x) = 1, we will simplify the expression step by step.
Step 1: Rewrite the cotangent function in terms of sine and cosine.
cot(x) = cos(x) / sin(x),
cot(2x) = cos(2x) / sin(2x),
cot(3x) = cos(3x) / sin(3x).
Now, let's substitute these values into the equation:
cos(x) / sin(x) * cos(2x) / sin(2x) - cos(2x) / sin(2x) * cos(3x) / sin(3x) - cos(3x) / sin(3x) * cos(x) / sin(x) = 1.
Step 2: Simplify the expression using trigonometric identities.
First, we'll try to find a common denominator for all the fractions. The common denominator will be sin(x) * sin(2x) * sin(3x).
(cos(x) * cos(2x) * sin(2x) * sin(3x) - cos(2x) * cos(3x) * sin(x) * sin(3x) - cos(3x) * cos(x) * sin(2x) * sin(x))/(sin(x) * sin(2x) * sin(3x)) = 1.
Now, let's simplify the numerator further.
(cos(x) * cos(2x) * sin(2x) * sin(3x) - cos(2x) * cos(3x) * sin(x) * sin(3x) - cos(3x) * cos(x) * sin(2x) * sin(x)) = sin(x) * sin(2x) * sin(3x).
Step 3: Expand and simplify the trigonometric terms using product-to-sum identities.
Expanding and rearranging the terms, we get:
(cos(x) * (cos(x) * cos(2x) * sin(2x) - cos(3x) * sin(x))) + (-sin(x) * (cos(2x) * cos(3x) * sin(3x) - cos(x) * sin(2x)))) = sin(x) * sin(2x) * sin(3x).
Using the product-to-sum identities:
(cos(x) * (cos(x) * (2 * cos(x) * sin(x)) - (4 * cos(x)^3 - 3 * cos(x) * sin(x)))) + (-sin(x) * (cos(2x) * (4 * cos(x)^2 - sin(x)^2) - (2 * cos(x) * sin(x) * (cos(2x) - cos(x)))))) = sin(x) * sin(2x) * sin(3x).
Step 4: Continue simplifying and expanding the terms.
Simplifying further, we get:
2 * cos(x)^3 - 3 * cos(x) * sin(x) + 4 * cos(x)^3 * cos(2x) - 4 * cos(x)^2 * sin(x)^2 - 2 * cos(x) * sin(x) * cos(3x) + 2 * cos(x) * sin(x) * cos(2x) = sin(x) * sin(2x) * sin(3x).
Expanding and rearranging the terms:
6 * cos(x)^3 - 5 * cos(x) * sin(x) - 4 * cos(x)^2 * sin(x)^2 - 2 * cos(x) * sin(x) * (cos(3x) - cos(2x)) = sin(x) * sin(2x) * sin(3x).
Step 5: Simplify and solve the equation.
After simplifying, we have:
(6 * cos(x)^3 - 5 * cos(x) * sin(x) - 4 * cos(x)^2 * sin(x)^2) - 2 * cos(x) * sin(x) * (cos(3x) - cos(2x)) = sin(x) * sin(2x) * sin(3x).
At this stage, it is challenging to further simplify the equation or find exact values for x. You may consider using numerical approximation methods or graphing the equation to estimate possible solutions.