Simplify cos(a+b)cos(a-b)

Problem

cos(a+b)*cos(a−b)

Solution

1. Apply the product-to-sum formula for the product of two cosines, which states cos(x)*cos(y)=1/2*(cos(x−y)+cos(x+y))

2. Substitute x=a+b and y=a−b into the formula.

3. Simplify the arguments of the cosine functions by performing the addition and subtraction.

x−y=(a+b)−(a−b)=2*b

x+y=(a+b)+(a−b)=2*a

4. Rewrite the expression using the simplified arguments.

1/2*(cos(2*b)+cos(2*a))

5. Apply double angle identities cos(2*a)=2*cos2(a)−1 and cos(2*b)=1−2*sin2(b) to express the result in terms of a and b

1/2*(1−2*sin2(b)+2*cos2(a)−1)

6. Simplify the constants and distribute the fraction.

1/2*(2*cos2(a)−2*sin2(b))

cos2(a)−sin2(b)

Final Answer

cos(a+b)*cos(a−b)=cos2(a)−sin2(b)

Want more problems? Check here!