Verify the Identity csc(x)^2(1-cos(x)^2)=1

Problem

csc2(x)*(1−cos2(x))=1

Solution

1. Identify the Pythagorean identity for sine and cosine, which states that sin2(x)+cos2(x)=1

2. Rearrange the identity to find an expression for 1−cos2(x)

sin2(x)=1−cos2(x)

3. Substitute the expression sin2(x) into the left side of the original equation.

csc2(x)*sin2(x)=1

4. Apply the reciprocal identity for cosecant, which is csc(x)=1/sin(x)

csc2(x)=1/sin2(x)

5. Substitute the reciprocal identity into the equation.

1/sin2(x)*sin2(x)=1

6. Simplify the left side by canceling the sin2(x) terms.

1=1

Final Answer

csc2(x)*(1−cos2(x))=1

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