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

Problem

sec(x)+tan(x)=cos(x)/(1−sin(x))

Solution

1. Rewrite the left side of the equation using the fundamental trigonometric identities for secant and tangent.

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

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

1/cos(x)+sin(x)/cos(x)=(1+sin(x))/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))/cos(x)⋅(1−sin(x))/(1−sin(x))=(1−sin2(x))/(cos(x)*(1−sin(x)))

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

(1−sin2(x))/(cos(x)*(1−sin(x)))=cos2(x)/(cos(x)*(1−sin(x)))

5. Cancel the common factor of cos(x) from the numerator and the denominator to reach the desired form.

cos2(x)/(cos(x)*(1−sin(x)))=cos(x)/(1−sin(x))

Final Answer

sec(x)+tan(x)=cos(x)/(1−sin(x))

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