simplify to a constant or basic trig function..

1+tan(x)/1+cot(x)

To simplify the expression 1 + tan(x)/1 + cot(x), let's break it down step by step.

1. Start by simplifying tan(x) and cot(x) individually. Remember that tan(x) is equal to sin(x)/cos(x) and cot(x) is equal to cos(x)/sin(x).

tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

2. Substitute the values of tan(x) and cot(x) back into the original expression.

1 + (sin(x)/cos(x)) / 1 + (cos(x)/sin(x))

3. Next, simplify the expression by getting a common denominator for the fractions in the numerator and denominator.

To get a common denominator, multiply the numerator and denominator of the first fraction by sin(x) and multiply the numerator and denominator of the second fraction by cos(x).

(sin(x) + cos(x)) / sin(x)cos(x) + cos^2(x) /sin(x)

4. Combine the fractions.

(sin(x) + cos(x) + cos^2(x)) / sin(x)cos(x) + cos^2(x)

5. Simplify the expression further by factoring out common terms.

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

6. The cos^2(x) can be further simplified using the identity cos^2(x) = 1 - sin^2(x). Replace cos^2(x) in the numerator with 1 - sin^2(x).

(1 + cos(x) + 1 - sin^2(x)) / cos(x)(sin(x) + cos(x))

7. Combine like terms in the numerator.

(2 + cos(x) - sin^2(x)) / cos(x)(sin(x) + cos(x))

8. Use the identity sin^2(x) = 1 - cos^2(x) to simplify further.

(2 + cos(x) - (1 - cos^2(x))) / cos(x)(sin(x) + cos(x))

9. Expand the equation in the numerator.

(2 + cos(x) - 1 + cos^2(x)) / cos(x)(sin(x) + cos(x))

10. Combine like terms in the numerator.

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

11. Divide both the numerator and denominator by cos(x).

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

After following these steps, you have simplified the expression 1 + tan(x)/1 + cot(x) to (cos^2(x) + cos(x) + 1) / (sin(x) + cos(x)).