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

Problem

(sin2(x)−1)/cos(−x)

Solution

1. Apply the Pythagorean identity to the numerator by recognizing that sin2(x)+cos2(x)=1 which implies sin2(x)−1=−cos2(x)

sin2(x)−1=−cos2(x)

2. Apply the even-odd identity for the cosine function, which states that cos(−x)=cos(x)

cos(−x)=cos(x)

3. Substitute these expressions back into the original fraction.

(−cos2(x))/cos(x)

4. Simplify the fraction by dividing the numerator by the denominator.

(−cos2(x))/cos(x)=−cos(x)

Final Answer

(sin2(x)−1)/cos(−x)=−cos(x)

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