Verify the Identity

Problem

1/tan2(x)−1/cot2(x)=csc2(x)−sec2(x)

Solution

1. Use reciprocal identities to rewrite the terms on the left side of the equation.

1/tan2(x)=cot2(x)

1/cot2(x)=tan2(x)

2. Substitute these expressions back into the left side of the identity.

cot2(x)−tan2(x)

3. Apply Pythagorean identities to relate the cotangent and tangent functions to cosecant and secant.

cot2(x)=csc2(x)−1

tan2(x)=sec2(x)−1

4. Substitute the Pythagorean identities into the expression.

(csc2(x)−1)−(sec2(x)−1)

5. Distribute the negative sign and simplify the expression by canceling the constant terms.

csc2(x)−1−sec2(x)+1

csc2(x)−sec2(x)

Final Answer

1/tan2(x)−1/cot2(x)=csc2(x)−sec2(x)

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