Verify the Identity 1-(cos(x)^2)/(1+sin(x))=sin(x)

Problem

1−cos2(x)/(1+sin(x))=sin(x)

Solution

1. Identify the Pythagorean identity for the numerator cos2(x)

cos2(x)=1−sin2(x)

2. Substitute the identity into the left side of the equation.

1−(1−sin2(x))/(1+sin(x))

3. Factor the numerator 1−sin2(x) as a difference of squares.

1−((1−sin(x))*(1+sin(x)))/(1+sin(x))

4. Simplify the fraction by canceling the common factor (1+sin(x)) from the numerator and denominator.

1−(1−sin(x))

5. Distribute the negative sign and combine like terms.

1−1+sin(x)

sin(x)

Final Answer

1−cos2(x)/(1+sin(x))=sin(x)

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