Simplify (1-sin(x))/(1+sin(x))

Problem

(1−sin(x))/(1+sin(x))

Solution

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

(1−sin(x))/(1+sin(x))⋅(1−sin(x))/(1−sin(x))

2. Expand the numerator and the denominator.

((1−sin(x))2)/(1−sin2(x))

3. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to the denominator.

((1−sin(x))2)/cos2(x)

4. Rewrite the expression as a single square.

((1−sin(x))/cos(x))2

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

(1/cos(x)−sin(x)/cos(x))2

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

(sec(x)−tan(x))2

Final Answer

(1−sin(x))/(1+sin(x))=(sec(x)−tan(x))2

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