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

Problem

cos(x+y)*cos(x−y)=cos^2(x)−sin^2(y)

Solution

1. Apply the sum and difference formulas for cosine to expand the left side of the equation.

cos(x+y)=cos(x)*cos(y)−sin(x)*sin(y)

cos(x−y)=cos(x)*cos(y)+sin(x)*sin(y)

2. Multiply the two expressions using the difference of squares pattern, (a−b)*(a+b)=a^2−b^2

cos(x+y)*cos(x−y)=(cos(x)*cos(y))^2−(sin(x)*sin(y))^2

cos(x+y)*cos(x−y)=cos^2(x)*cos^2(y)−sin^2(x)*sin^2(y)

3. Use Pythagorean identities to replace cos^2(y) and sin^2(x) so the expression only contains cos^2(x) and sin^2(y)

cos^2(y)=1−sin^2(y)

sin^2(x)=1−cos^2(x)

4. Substitute these identities back into the expanded expression.

cos^2(x)*(1−sin^2(y))−(1−cos^2(x))*sin^2(y)

5. Distribute and simplify the terms to reach the final form.

cos^2(x)−cos^2(x)*sin^2(y)−(sin^2(y)−cos^2(x)*sin^2(y))

cos^2(x)−cos^2(x)*sin^2(y)−sin^2(y)+cos^2(x)*sin^2(y)

cos^2(x)−sin^2(y)

Final Answer

cos(x+y)*cos(x−y)=cos^2(x)−sin^2(y)

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