Proof:

To prove the given equation, we can use the trigonometric identities and properties.

Trigonometric Identities:

1. Sine Double Angle Identity: sin(2θ) = 2sin(θ)cos(θ)
2. Sine Triple Angle Identity: sin(3θ) = 3sin(θ) - 4sin^3(θ)

Given Equation:

sin(A)sin(60-A)sin(60+A) = 1/4sin(3A)

Proof:

Let's start by simplifying the left side of the given equation.

sin(A)sin(60-A)sin(60+A)

Using the product-to-sum identity for sine: sin(θ)sin(φ) = (1/2)(cos(θ-φ) - cos(θ+φ))

sin(A)sin(60-A)sin(60+A) = (1/2)(cos(60-2A) - cos(120))

We know that cos(60) = 1/2, so we can rewrite the equation as:

(1/2)(cos(60-2A) - cos(120)) = (1/2)(cos(60-2A) + cos(60))

Using the sum-to-product identity for cosine: cos(θ) + cos(φ) = 2cos((θ+φ)/2)cos((θ-φ)/2)

(1/2)(cos(60-2A) + cos(60)) = (1/2)(2cos((60-2A+60)/2)cos((60-2A-60)/2))

Simplifying further:

(1/2)(2cos(60-A)cos(-A)) = cos(60-A)cos(A)

Now, let's simplify the right side of the given equation.

1/4sin(3A)

Using the sine triple angle identity: sin(3θ) = 3sin(θ) - 4sin^3(θ)

1/4(3sin(A) - 4sin^3(A))

Expanding and simplifying:

3/4sin(A) - sin^3(A)

Comparing the left side and the right side of the given equation, we have:

cos(60-A)cos(A) = 3/4sin(A) - sin^3(A)

Using the identity sin^2(A) = 1 - cos^2(A), we can rewrite the equation as:

cos(60-A)cos(A) = 3/4sin(A) - (1 - cos^2(A))sin(A)

Expanding and simplifying:

cos(60-A)cos(A) = 3/4sin(A) - sin(A) + cos^2(A)sin(A)

Rearranging the terms:

cos(60-A)cos(A) + sin(A) - cos^2(A)sin(A) = 3/4sin(A)

Using the identity sin(A) = sin(180-A), we can rewrite the equation as:

cos(60-A)cos(A) + sin(180-A) - cos^2(A)sin(180-A) = 3/4sin(A)

Expanding and simplifying:

cos(60