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

Problem

(1−cos(x))/(1+cos(x))

Solution

1. Identify the half-angle identities for sine and cosine, which relate 1−cos(x) and 1+cos(x) to squared trigonometric functions.

2. Substitute the identity 1−cos(x)=2*sin2(x/2) into the numerator.

3. Substitute the identity 1+cos(x)=2*cos2(x/2) into the denominator.

4. Simplify the expression by canceling the common factor of 2

5. Apply the definition of the tangent function, where sin(θ)/cos(θ)=tan(θ) to the squared terms.

Final Answer

(1−cos(x))/(1+cos(x))=tan2(x/2)

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