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

Problem

tan(x)+sec(x)=cos(x)/(1−sin(x))

Solution

1. Rewrite the left side of the identity using sine and cosine functions.

tan(x)+sec(x)=sin(x)/cos(x)+1/cos(x)

2. Combine the fractions since they share a common denominator.

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

3. Multiply the numerator and the denominator by the conjugate of the numerator, which is 1−sin(x) to transform the expression.

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

4. Expand the numerator using the difference of squares formula.

(1−sin2(x))/(cos(x)*(1−sin(x)))

5. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to simplify the numerator.

cos2(x)/(cos(x)*(1−sin(x)))

6. Cancel the common factor of cos(x) from the numerator and the denominator.

cos(x)/(1−sin(x))

Final Answer

tan(x)+sec(x)=cos(x)/(1−sin(x))

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