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

Problem

sec(y)+tan(y)=cos(y)/(1−sin(y))

Solution

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

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

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

1/cos(y)+sin(y)/cos(y)=(1+sin(y))/cos(y)

3. Multiply the numerator and the denominator by the conjugate of the numerator, which is 1−sin(y)

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

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

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

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

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

Final Answer

sec(y)+tan(y)=cos(y)/(1−sin(y))

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