Verify the Identity 4csc(2x)=2csc(x)^2tan(x)

Problem

4*csc(2*x)=2*csc^2(x)*tan(x)

Solution

1. Express the left side using the reciprocal identity for cosecant.

4*csc(2*x)=4/sin(2*x)

2. Apply the double-angle formula for sine, sin(2*x)=2*sin(x)*cos(x) to the denominator.

4/sin(2*x)=4/(2*sin(x)*cos(x))

3. Simplify the fraction by dividing the numerator and denominator by 2.

4/(2*sin(x)*cos(x))=2/(sin(x)*cos(x))

4. Express the right side using the definitions of cosecant and tangent in terms of sine and cosine.

2*csc^2(x)*tan(x)=2⋅1/sin^2(x)⋅sin(x)/cos(x)

5. Simplify the right side by canceling one factor of sin(x) from the numerator and denominator.

2⋅1/sin^2(x)⋅sin(x)/cos(x)=(2*sin(x))/(sin^2(x)*cos(x))

(2*sin(x))/(sin^2(x)*cos(x))=2/(sin(x)*cos(x))

6. Compare both sides to confirm they are equivalent.

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

Final Answer

4*csc(2*x)=2*csc^2(x)*tan(x)

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