Question

Question #c42b1

Answer 1

We will use the following:

• `cos(alpha+beta) = cos(alpha)cos(beta)-sin(alpha)sin(beta)`
• `cos(alpha - beta) = cos(alpha)cos(beta) + sin(alpha)sin(beta)`
• `cos(45^@) = sin(45^@) = sqrt(2)/2`

With those, we have

`cos(theta+45^@) = cos(theta)cos(45^@)-sin(theta)sin(45^@)`

`=sqrt(2)/2cos(theta)-sqrt(2)/2sin(theta)`

`cos(theta-45^@) = cos(theta)cos(45^@)+sin(theta)sin(45^@)`

`=sqrt(2)/2cos(theta)+sqrt(2)/2sin(theta)`

Adding the two, we get

`cos(theta+45^@)+cos(theta-45^@)`

`=sqrt(2)/2cos(theta)-sqrt(2)/2sin(theta)+sqrt(2)/2cos(theta)+sqrt(2)/2sin(theta)`

`=2(sqrt(2)/2cos(theta))`

`=sqrt(2)cos(theta)`