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

Problem

cos(x)/(1−sin(x))+(1−sin(x))/cos(x)

Solution

1. Find a common denominator by multiplying the first fraction by cos(x)/cos(x) and the second fraction by (1−sin(x))/(1−sin(x))

(cos2(x)+(1−sin(x))2)/(cos(x)*(1−sin(x)))

2. Expand the numerator by squaring the binomial (1−sin(x))2

(cos2(x)+1−2*sin(x)+sin2(x))/(cos(x)*(1−sin(x)))

3. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the numerator.

(1+1−2*sin(x))/(cos(x)*(1−sin(x)))

4. Combine like terms in the numerator.

(2−2*sin(x))/(cos(x)*(1−sin(x)))

5. Factor out the constant 2 from the numerator.

(2*(1−sin(x)))/(cos(x)*(1−sin(x)))

6. Cancel the common factor (1−sin(x)) from the numerator and denominator.

2/cos(x)

7. Apply the reciprocal identity 1/cos(x)=sec(x) to reach the final form.

2*sec(x)

Final Answer

cos(x)/(1−sin(x))+(1−sin(x))/cos(x)=2*sec(x)

Want more problems? Check here!