Simplify 1/(cos(x)^2)-1/(cot(x)^2)

Problem

1/cos2(x)−1/cot2(x)

Solution

1. Identify the reciprocal identities for the trigonometric functions in the expression.

2. Substitute the identity sec(x)=1/cos(x) into the first term.

1/cos2(x)=sec2(x)

3. Substitute the identity tan(x)=1/cot(x) into the second term.

1/cot2(x)=tan2(x)

4. Rewrite the original expression using these substitutions.

sec2(x)−tan2(x)

5. Apply the Pythagorean identity tan2(x)+1=sec2(x) which can be rearranged to sec2(x)−tan2(x)=1

sec2(x)−tan2(x)=1

Final Answer

1/cos2(x)−1/cot2(x)=1

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