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

Problem

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

Solution

1. Apply half-angle identities to rewrite the numerator and denominator in terms of x/2

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

3. Substitute the double-angle identity sin(x)=2*sin(x/2)*cos(x/2) into the denominator.

4. Simplify the fraction by canceling the common factor 2*cos(x/2) from both the numerator and the denominator.

5. Identify the resulting trigonometric ratio using the definition cot(θ)=cos(θ)/sin(θ)

Final Answer

(1+cos(x))/sin(x)=cot(x/2)

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