Simplify (cos(x))/(1-tan(x))+(sin(x))/(1-cot(x))

Problem

cos(x)/(1−tan(x))+sin(x)/(1−cot(x))

Solution

1. Rewrite the tangent and cotangent functions in terms of sine and cosine.

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

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

2. Substitute these expressions into the original denominators.

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

3. Find a common denominator for the terms within each denominator.

cos(x)/(cos(x)−sin(x))/cos(x)+sin(x)/(sin(x)−cos(x))/sin(x)

4. Simplify the complex fractions by multiplying the numerators by the reciprocals of the denominators.

cos2(x)/(cos(x)−sin(x))+sin2(x)/(sin(x)−cos(x))

5. Adjust the sign of the second fraction to create a common denominator.

cos2(x)/(cos(x)−sin(x))−sin2(x)/(cos(x)−sin(x))

6. Combine the fractions over the common denominator.

(cos2(x)−sin2(x))/(cos(x)−sin(x))

7. Factor the numerator using the difference of squares formula.

((cos(x)−sin(x))*(cos(x)+sin(x)))/(cos(x)−sin(x))

8. Cancel the common factor in the numerator and denominator.

cos(x)+sin(x)

Final Answer

cos(x)/(1−tan(x))+sin(x)/(1−cot(x))=cos(x)+sin(x)

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