Verify the Identity

Problem

(csc(6*x)−cot(6*x))/(csc(6*x)+cot(6*x))=tan2(3*x)

Solution

1. Rewrite the left-hand side using the definitions of cosecant and cotangent in terms of sine and cosine.

(1/sin(6*x)−cos(6*x)/sin(6*x))/(1/sin(6*x)+cos(6*x)/sin(6*x))

2. Simplify the complex fraction by multiplying the numerator and denominator by sin(6*x)

(1−cos(6*x))/(1+cos(6*x))

3. Apply the half-angle identities for sine and cosine, specifically 1−cos(θ)=2*sin2(θ/2) and 1+cos(θ)=2*cos2(θ/2) where θ=6*x

(2*sin2(3*x))/(2*cos2(3*x))

4. Cancel the common factor of 2 and use the definition of tangent, tan(θ)=sin(θ)/cos(θ)

tan2(3*x)

Final Answer

(csc(6*x)−cot(6*x))/(csc(6*x)+cot(6*x))=tan2(3*x)

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