Verify the Identity cos(x)^2(1+tan(x)^2)=1

Problem

cos2(x)*(1+tan2(x))=1

Solution

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

2. Substitute the identity 1+tan2(x)=sec2(x) into the left side of the equation.

cos2(x)*sec2(x)

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

cos2(x)*(1/cos(x))2

4. Simplify the expression by squaring the fraction.

cos2(x)1/cos2(x)

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

1

Final Answer

cos2(x)*(1+tan2(x))=1

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