Simplify square root of (1+sin(y))/(1-sin(y))

Problem

√(,(1+sin(y))/(1−sin(y)))

Solution

1. Multiply the numerator and the denominator inside the square root by the conjugate of the denominator, which is 1+sin(y)

√(,((1+sin(y))*(1+sin(y)))/((1−sin(y))*(1+sin(y))))

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

√(,((1+sin(y))2)/(1−sin2(y)))

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

√(,((1+sin(y))2)/cos2(y))

4. Simplify the square root by taking the square root of the numerator and the denominator separately.

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

5. Distribute the denominator to each term in the numerator.

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

6. Use trigonometric identities sec(y)=1/cos(y) and tan(y)=sin(y)/cos(y) to reach the final form.

sec(y)+tan(y)

Final Answer

√(,(1+sin(y))/(1−sin(y)))=sec(y)+tan(y)

Want more problems? Check here!