Trigonometric Identity:

To prove the trigonometric identity cot(x/2) * cot(y/2) * cot(z/2) = cot(x/2) * cot(y/2) * cot(z/2), given that x + y + z = π, we need to apply trigonometric identities and properties to simplify the expression on both sides of the equation.

Using the Sum-to-Product Identities:

First, let's rewrite the given equation in terms of sine and cosine using the sum-to-product identities:
cot(x/2) * cot(y/2) * cot(z/2) = cot(x/2) * cot(y/2) * cot(z/2)

Since x + y + z = π, we can rewrite the equation as:
cot(x/2) * cot(y/2) * cot(z/2) = cot(x/2) * cot(y/2) * cot(π - x/2 - y/2)

Using the Cotangent Identity:

Next, we can apply the cotangent identity cot(π - θ) = -cot(θ) to simplify the expression on the right-hand side:
cot(x/2) * cot(y/2) * cot(z/2) = cot(x/2) * cot(y/2) * (-cot(x/2 + y/2))

Applying the Product-to-Sum Identity:

To further simplify the expression, we can use the product-to-sum identity cot(A) * cot(B) = (cot(A) + cot(B)) / (cot(A) - cot(B)). Applying this identity to the right-hand side of the equation, we get:
cot(x/2) * cot(y/2) * cot(z/2) = cot(x/2) * cot(y/2) * (-(cot(x/2) + cot(y/2)) / (cot(x/2) - cot(y/2)))

Cancelling out the Cotangent Terms:

Now, we can cancel out the cot(x/2) * cot(y/2) terms on both sides of the equation, leaving us with:
cot(z/2) = -(cot(x/2) + cot(y/2)) / (cot(x/2) - cot(y/2))

Manipulating the Right-Hand Side:

To simplify the right-hand side further, we can multiply both the numerator and denominator by (cot(x/2) - cot(y/2)):
cot(z/2) = -((cot(x/2) + cot(y/2)) * (cot(x/2) - cot(y/2))) / ((cot(x/2) - cot(y/2)) * (cot(x/2) - cot(y/2)))

Using the Difference of Squares Identity:

Now, we can apply the difference of squares identity (a^2 - b^2) = (a + b)(a - b) to simplify the expression on the right-hand side:
cot(z/2) = -((cot(x/2))^2 - (cot(y/2))^2) / ((cot(x/2) - cot(y/2))^2)

Using the Pythagorean Identity:

Finally, we can apply the