Expand the Trigonometric Expression (sin(2x))/(1-cos(2x))

Problem

sin(2*x)/(1−cos(2*x))

Solution

1. Apply the double angle identity for the numerator using sin(2*x)=2*sin(x)*cos(x)

2. Apply the double angle identity for the denominator using cos(2*x)=1−2*sin2(x)

3. Substitute the identity into the denominator expression 1−cos(2*x)

1−(1−2*sin2(x))

4. Simplify the denominator by distributing the negative sign.

1−1+2*sin2(x)=2*sin2(x)

5. Rewrite the entire fraction with the substituted identities.

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

6. Cancel the common factors of 2 and sin(x) from the numerator and denominator.

cos(x)/sin(x)

7. Identify the resulting expression as the cotangent identity.

cot(x)

Final Answer

sin(2*x)/(1−cos(2*x))=cot(x)

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