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

Problem

(1+sin(x))*(1−sin(x))=cos^2(x)

Solution

1. Expand the left side of the equation using the difference of squares formula, (a+b)*(a−b)=a^2−b^2

(1+sin(x))*(1−sin(x))=1^2−sin^2(x)

2. Simplify the expression by evaluating the square of 1.

1^2−sin^2(x)=1−sin^2(x)

3. Apply the Pythagorean identity, sin^2(x)+cos^2(x)=1 which can be rearranged to show that 1−sin^2(x)=cos^2(x)

1−sin^2(x)=cos^2(x)

4. Conclude that the left side of the equation is equal to the right side, verifying the identity.

cos^2(x)=cos^2(x)

Final Answer

(1+sin(x))*(1−sin(x))=cos^2(x)

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