Simplify and write the trigonometric expression in terms of sine and cosine:

sin x + (cot x cos x) = 1/(f(x)
f(x)= ???

sin x + (cos^2 x /sin x) = 1/y

sin^2 x/sin^2 x + cos^2 x/sin^2 x = 1/y

1/sin^2 x = 1/y

y = sin^2 x

To simplify and write the trigonometric expression in terms of sine and cosine, we can first simplify the cotangent function.

Recall that cot(x) is defined as the reciprocal of tangent, so cot(x) = 1/tan(x). And tangent is defined as sine divided by cosine, so tan(x) = sin(x)/cos(x).

Substituting this back into the original expression, we have:

sin(x) + (1/tan(x)) * cos(x) = 1/f(x)

Substituting the definition of tangent, we get:

sin(x) + (1/(sin(x)/cos(x))) * cos(x) = 1/f(x)

Now, let's simplify further:

sin(x) + (cos(x)/sin(x)) * cos(x) = 1/f(x)

Multiplying the two fractions together:

sin(x) + (cos^2(x))/sin(x) = 1/f(x)

To add these fractions together, we need a common denominator. The common denominator is sin(x), so we multiply the first fraction by cos(x)/cos(x) to get a common denominator:

(sinx * cos(x)/cos(x)) + (cos^2(x))/sin(x) = 1/f(x)

Simplifying:

(sin(x)*cos(x) + cos^2(x))/sin(x) = 1/f(x)

Now, let's simplify the numerator:

sin(x)*cos(x) + cos^2(x) = cos(x)*(sin(x) + cos(x))

So the expression becomes:

(cos(x)*(sin(x) + cos(x)))/sin(x) = 1/f(x)

And now we can simplify the expression by canceling out the sin(x) term:

(cos(x) * (1 + cos(x)/sin(x))) = 1/f(x)

Finally, we can use the identity sin^2(x) + cos^2(x) = 1 to simplify the fraction term:

(cos(x) * (1 + cos(x)/sqrt(1 - cos^2(x)))) = 1/f(x)

Therefore, f(x) = cos(x) * (1 + cos(x)/sqrt(1 - cos^2(x)))