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

Problem

sin(x)/(1+cos(x))=(1−cos(x))/sin(x)

Solution

1. Identify the left side of the equation to begin the verification process.

LHS=sin(x)/(1+cos(x))

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

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

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

(sin(x)*(1−cos(x)))/(1−cos2(x))

4. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

(sin(x)*(1−cos(x)))/sin2(x)

5. Simplify the expression by canceling one factor of sin(x) from the numerator and the denominator.

(1−cos(x))/sin(x)

6. Compare the result to the right side of the original identity.

(1−cos(x))/sin(x)=RHS

Final Answer

sin(x)/(1+cos(x))=(1−cos(x))/sin(x)

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