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

Problem

1+sec2(x)*sin2(x)=sec2(x)

Solution

1. Identify the trigonometric identity for the secant function, which is sec(x)=1/cos(x)

2. Substitute the identity into the left side of the equation to express the term in terms of sine and cosine.

1+(1/cos2(x))*sin2(x)

3. Multiply the fractions to combine the terms into a single trigonometric ratio.

1+sin2(x)/cos2(x)

4. Apply the tangent identity tan(x)=sin(x)/cos(x) to simplify the expression.

1+tan2(x)

5. Use the Pythagorean identity 1+tan2(x)=sec2(x) to reach the final form.

sec2(x)

Final Answer

1+sec2(x)*sin2(x)=sec2(x)

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