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

Problem

cos(x+y)*cos(x−y)=cos2(x)−sin2(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)=a2−b2

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

cos(x+y)*cos(x−y)=cos2(x)*cos2(y)−sin2(x)*sin2(y)

3. Use Pythagorean identities to replace cos2(y) and sin2(x) so the expression only contains cos2(x) and sin2(y)

cos2(y)=1−sin2(y)

sin2(x)=1−cos2(x)

4. Substitute these identities back into the expanded expression.

cos2(x)*(1−sin2(y))−(1−cos2(x))*sin2(y)

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

cos2(x)−cos2(x)*sin2(y)−(sin2(y)−cos2(x)*sin2(y))

cos2(x)−cos2(x)*sin2(y)−sin2(y)+cos2(x)*sin2(y)

cos2(x)−sin2(y)

Final Answer

cos(x+y)*cos(x−y)=cos2(x)−sin2(y)

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