Verify the Identity (cos(x))/(1-sin(x))=sec(x)+tan(x)

Problem

cos(x)/(1−sin(x))=sec(x)+tan(x)

Solution

1. Multiply the numerator and the denominator of the left side by the conjugate of the denominator, which is 1+sin(x)

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

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

(cos(x)*(1+sin(x)))/(1−sin2(x))

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

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

4. Simplify the fraction by canceling one factor of cos(x) from the numerator and the denominator.

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

5. Split the fraction into two separate terms.

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

6. Substitute the reciprocal and quotient identities sec(x)=1/cos(x) and tan(x)=sin(x)/cos(x)

sec(x)+tan(x)

Final Answer

cos(x)/(1−sin(x))=sec(x)+tan(x)

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