Simplify (sec(x)^2-tan(x)^2)/(cot(x)^2-csc(x)^2)

Problem

(sec2(x)−tan2(x))/(cot2(x)−csc2(x))

Solution

1. Identify the Pythagorean identity for secant and tangent, which states 1+tan2(x)=sec2(x)

2. Rearrange the identity to find the value of the numerator: sec2(x)−tan2(x)=1

3. Identify the Pythagorean identity for cosecant and cotangent, which states 1+cot2(x)=csc2(x)

4. Rearrange the identity to find the value of the denominator: cot2(x)−csc2(x)=−1

5. Substitute these values back into the original fraction to simplify the expression.

6. Divide the resulting constants to find the final value.

Final Answer

(sec2(x)−tan2(x))/(cot2(x)−csc2(x))=−1

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