Verify the Identity tan(x/2)=csc(x)-cot(x)

Problem

tan(x/2)=csc(x)−cot(x)

Solution

1. Apply the half-angle identity for the tangent function, specifically the version that relates to sine and cosine.

tan(x/2)=(1−cos(x))/sin(x)

2. Split the fraction into two separate terms by dividing each term in the numerator by the common denominator.

(1−cos(x))/sin(x)=1/sin(x)−cos(x)/sin(x)

3. Use reciprocal and quotient identities to rewrite the terms in terms of cosecant and cotangent.

1/sin(x)=csc(x)

cos(x)/sin(x)=cot(x)

4. Substitute the identities back into the expression to reach the right-hand side of the original equation.

csc(x)−cot(x)

Final Answer

tan(x/2)=csc(x)−cot(x)

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