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

Problem

cos(x+y)+cos(x−y)=2*cos(x)*cos(y)

Solution

1. Apply the cosine sum identity to the first term cos(x+y)

cos(x+y)=cos(x)*cos(y)−sin(x)*sin(y)

2. Apply the cosine difference identity to the second term cos(x−y)

cos(x−y)=cos(x)*cos(y)+sin(x)*sin(y)

3. Substitute these expansions back into the left side of the original equation.

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

4. Combine like terms by noticing that −sin(x)*sin(y) and +sin(x)*sin(y) cancel each other out.

cos(x)*cos(y)+cos(x)*cos(y)

5. Simplify the expression to reach the right side of the identity.

2*cos(x)*cos(y)

Final Answer

cos(x+y)+cos(x−y)=2*cos(x)*cos(y)

Want more problems? Check here!