The given differential equation is:
cos(x) * dy/dx * y * sin(x) = 1

We can rewrite the equation as:
dy/dx = 1 / (y * cos(x) * sin(x))

Let's find the integrating factor for this differential equation:

Step 1: Identify the form of the differential equation
The given differential equation is not in standard form, but it can be transformed into a linear differential equation by multiplying both sides by cos(x) * sin(x):

cos(x) * sin(x) * dy/dx * y = 1 * cos(x) * sin(x)

This gives us the standard form of a linear differential equation:

dy/dx * y = cos(x) * sin(x)

Step 2: Identify the coefficient of y
The coefficient of y in the standard form is 1.

Step 3: Find the integrating factor
The integrating factor (IF) is given by the formula:

IF = exp ∫P(x) dx

Where P(x) is the coefficient of y.

In this case, P(x) = 1, so the integrating factor is:

IF = exp ∫1 dx

IF = exp(x)

Therefore, the integrating factor for the given differential equation is exp(x), or simply e^x.

Step 4: Multiply both sides of the differential equation by the integrating factor
Multiplying both sides of the differential equation by the integrating factor e^x, we get:

e^x * dy/dx * y = e^x * cos(x) * sin(x)

The left side of the equation can be written as the derivative of (e^x * y) with respect to x:

d/dx (e^x * y) = e^x * cos(x) * sin(x)

Step 5: Integrate both sides of the equation
Integrating both sides of the equation with respect to x, we get:

∫ d/dx (e^x * y) dx = ∫ e^x * cos(x) * sin(x) dx

e^x * y = ∫ e^x * cos(x) * sin(x) dx

Step 6: Solve for y
To solve for y, we divide both sides of the equation by e^x:

y = (∫ e^x * cos(x) * sin(x) dx) / e^x

Simplifying the integral and canceling out e^x, we get:

y = ∫ cos(x) * sin(x) dx

Using the trigonometric identity sin(2x) = 2sin(x)cos(x), we can rewrite the integral as:

y = ∫ (1/2) sin(2x) dx

Integrating, we get:

y = (-1/4) cos(2x) + C

Where C is the constant of integration.

Therefore, the solution to the given differential equation is:

y = (-1/4) cos(2x) + C

Conclusion:
The integrating factor of the given differential equation cos(x) * dy/dx * y * sin(x) = 1 is e^x.