Proof:

To prove the given equation:

(cos(5x) * cos(4x))/(2cos(3x) - 1) = cos(2x) * cos(x)

Step 1: Simplify the left side of the equation

Let's start by simplifying the left side of the equation:

(cos(5x) * cos(4x))/(2cos(3x) - 1)

Step 2: Apply the angle addition formula for cosine

The angle addition formula for cosine states that:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Using this formula, we can rewrite cos(5x) and cos(4x) in terms of cos and sin:

cos(5x) = cos(3x + 2x) = cos(3x)cos(2x) - sin(3x)sin(2x) (1)

cos(4x) = cos(3x + x) = cos(3x)cos(x) - sin(3x)sin(x) (2)

Step 3: Substitute equations (1) and (2) into the left side of the equation

Substituting equations (1) and (2) into the left side of the equation, we get:

((cos(3x)cos(2x) - sin(3x)sin(2x)) * (cos(3x)cos(x) - sin(3x)sin(x)))/(2cos(3x) - 1)

Step 4: Simplify the numerator

Expanding the numerator, we have:

(cos(3x)cos(2x)cos(3x)cos(x) - cos(3x)cos(2x)sin(3x)sin(x) - sin(3x)sin(2x)cos(3x)cos(x) + sin(3x)sin(2x)sin(3x)sin(x))

Step 5: Simplify the denominator

Simplifying the denominator, we have:

2cos(3x) - 1

Step 6: Combine like terms

Combining like terms in the numerator, we get:

cos(3x)cos(3x)cos(2x)cos(x) - cos(3x)cos(2x)sin(3x)sin(x) - sin(3x)sin(2x)cos(3x)cos(x) + sin(3x)sin(2x)sin(3x)sin(x)

Step 7: Use the trigonometric identity

The trigonometric identity we will use is:

cos(2x) = 2cos^2(x) - 1

Using this identity, we can rewrite the equation:

(cos(3x)cos(3x)cos(2x)cos(x) - cos(3x)cos(2x)sin(3x)sin(x) - sin(3x)