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

Problem

(1−cos(x))/sin(x)=sin(x)/(1+cos(x))

Solution

1. Identify the starting side of the identity. We will begin with the left-hand side (LHS).

LHS=(1−cos(x))/sin(x)

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

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

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a^2−b^2

(1−cos^2(x))/(sin(x)*(1+cos(x)))

4. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 which implies 1−cos^2(x)=sin^2(x)

sin^2(x)/(sin(x)*(1+cos(x)))

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

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

6. Compare the result to the right-hand side (RHS) to confirm they are equal.

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

Final Answer

(1−cos(x))/sin(x)=sin(x)/(1+cos(x))

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