Simplify

Problem

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

Solution

1. Split the fractions by dividing each term in the numerators by their respective denominators.

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

2. Simplify the ratios using the definitions of trigonometric functions where sin(x)/cos(x)=tan(x) and cos(x)/sin(x)=cot(x)

1+tan(x)−(1−cot(x))

3. Distribute the negative sign through the parentheses.

1+tan(x)−1+cot(x)

4. Combine like terms by subtracting the constants.

tan(x)+cot(x)

5. Rewrite in terms of sine and cosine to see if further simplification is possible.

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

6. Find a common denominator to combine the fractions.

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

7. Apply the Pythagorean identity where sin2(x)+cos2(x)=1

1/(sin(x)*cos(x))

8. Use reciprocal identities where 1/sin(x)=csc(x) and 1/cos(x)=sec(x)

sec(x)*csc(x)

Final Answer

(cos(x)+sin(x))/cos(x)−(sin(x)−cos(x))/sin(x)=sec(x)*csc(x)

Want more problems? Check here!