Verify the Identity sec(x)^2(1-sin(x)^2)-1=0

Problem

sec2(x)*(1−sin2(x))−1=0

Solution

1. Identify the Pythagorean identity for the term inside the parentheses.

2. Substitute 1−sin2(x)=cos2(x) into the expression.

sec2(x)*cos2(x)−1=0

3. Apply the reciprocal identity for secant, where sec(x)=1/cos(x)

(1/cos(x))2*cos2(x)−1=0

4. Simplify the squared term.

1/cos2(x)*cos2(x)−1=0

5. Cancel the common factor of cos2(x) in the numerator and denominator.

1−1=0

6. Subtract the constants to verify the equality.

0=0

Final Answer

sec2(x)*(1−sin2(x))−1=0

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