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

Problem

(1−cos(x))/sin(x)

Solution

1. Identify the half-angle identities for the numerator and denominator to simplify the expression.

2. Apply the identity for the numerator using 1−cos(x)=2*sin2(x/2)

3. Apply the identity for the denominator using the double-angle formula sin(x)=2*sin(x/2)*cos(x/2)

4. Substitute these identities into the original fraction:

(2*sin2(x/2))/(2*sin(x/2)*cos(x/2))

5. Cancel the common factors of 2 and sin(x/2) from the numerator and denominator:

sin(x/2)/cos(x/2)

6. Simplify the resulting expression using the definition of the tangent function tan(θ)=sin(θ)/cos(θ)

Final Answer

(1−cos(x))/sin(x)=tan(x/2)

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