Find the exact value of the product cos 675 degree cos 225 d

Find the exact value of the product cos 67.5 degree cos 22.5 degree cos67.5 degree cos 22.5 degree =

Solution

We have given cos(67.5⁰)*cos(22.5⁰)

we know the formula for cosA*cosB=1/2*[cos(A+B)+cos(A-B)]

substitute A=67.5⁰ and B=22.5⁰

cos(67.5⁰)*cos(22.5⁰)=1/2*[cos(67.5⁰+22.5⁰)+cos(67.5⁰-22.5⁰)]

=1/2*[cos(90⁰)+cos(45⁰)]

=1/2*[0+1/sqrt(2)]

cos(67.5⁰)*cos(22.5⁰) =1/(2*sqrt(2))