Simplify the follow)ing to the simplest form

(tan(x) +cot(x))/ (tan(x)*cot(x))

I usually change everything to sines and cosines

(tan(x) +cot(x))/ (tan(x)*cot(x))
= (sinx/cosx + cosx/sinx) / (sinx/cosx * cosx/sinx)
= (sin^2 x + cos^2 x)/(sinxcosx) / 1
= 1/(sinxcosx) ----> I know 2sinxcosx = sin 2x
so sinxcosx = (1/2)sin2x

= 2/sin 2x or 2 csc 2x
or from the original:
1/(sinxcosx) = cscx secx

To simplify the expression "(tan(x) + cot(x))/(tan(x) * cot(x))" to its simplest form, we can use trigonometric identities.

Step 1: Identify the reciprocals and the Pythagorean identity.
- The reciprocal of the tangent function is the cotangent function: cot(x) = 1/tan(x).
- The reciprocal of the cotangent function is the tangent function: tan(x) = 1/cot(x).
- The Pythagorean identity for tangent is: tan^2(x) = 1 + cot^2(x).

Step 2: Substitute the reciprocal identities into the expression.
(tan(x) + cot(x))/(tan(x) * cot(x)) becomes ((1/cot(x)) + (cot(x))/(1/cot(x))) / ((1/cot(x)) * (cot(x))).

Step 3: Simplify the expression.
((1/cot(x)) + (cot(x))/(1/cot(x))) / ((1/cot(x)) * (cot(x)))
becomes ((1/cot(x)) + (cot^2(x))/(1)) / (1)
we can remove the denominators (cot(x)):
((1) + (cot^2(x))) / (1)
combining like terms gives us:
1 + cot^2(x)

Therefore, the simplified form of (tan(x) + cot(x))/(tan(x) * cot(x)) is 1 + cot^2(x).