R4 = 3( cos π/ 8 + cos π/ 4 + cos 3π /8 + cos π/ 2) π /8

Sorry, Your heading is different from what is in the body of the question.   ://

R4 = 3( cos π/ 8 + cos π/ 4 + cos 3π /8 + cos π/ 2) π /8

R4 = 3( cos π/ 8+ cos 3π /8 + 1/root2 ) π /8

R4 = 3( cos π/ 8+ sin π/ 8 + 1/root2 ) π /8

R4 = 3(sqrt(cos π/ 8+ sin π/ 8)^2  )   +1/root2 ) π /8

R4 = 3(sqrt(1+2sin(pi/8) cos(pi/8))   +1/root2 ) π /8

R4 = 3(sqrt(1+sin(pi/4) )   +1/root2 ) π /8

R4 = 3(sqrt(1+(1/root2 ))   +1/root2 ) π /8

I am sorry.  Trying to do this without LaTex is no fun at all.   :(

Maybe i will try to do more with it later.

OR perhaps you can take it from here.   

Your exact answer: R^4 = -304 pi

Your question does not make sense.

R4 = 3 (cos π 8 + cos π 4 + cos 3π 8 + cos π 2) π 8

Please mind your brackets. thanks.

The exact answer for this version is,

R^4 = -528 pi

A (not very edifying!) sum of cosines: 

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