Rationalize the Denominator (cos(x))/(1-sin(x))

Problem

cos(x)/(1−sin(x))

Solution

1. Identify the conjugate of the denominator 1−sin(x) which is 1+sin(x)

2. Multiply both the numerator and the denominator by this conjugate to eliminate the binomial in the denominator.

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

3. Expand the denominator using the difference of squares formula (a−b)*(a+b)=a2−b2

(cos(x)*(1+sin(x)))/(1−sin2(x))

4. Apply the Pythagorean identity cos2(x)+sin2(x)=1 which implies 1−sin2(x)=cos2(x)

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

5. Simplify the expression by canceling the common factor of cos(x) from the numerator and the denominator.

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

Final Answer

cos(x)/(1−sin(x))=(1+sin(x))/cos(x)

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