Prove that: `cos(A+B+C)+cos(A-B+C)+cos(A+B-C)+cos(-A+B+C)=4cos A\ cos B\ cos C`

Explanation

`cos (-A+pi-A)+cos (pi-B-B)+cos (pi-C-C)+cos pi`


`cos (pi-2 A)+cos (pi-2 B)+cos (pi-2 C)-1`


`-cos 2 A-cos 2 B-cos 2 C-1 quad cos (pi-theta)=-cos theta`


`-[(1+cos 2 B)+(cos 2 A+cos 2 C)] quad 1+cos ^{2} theta=2 cos ^{2} theta-1`


`-[2 cos ^{2} theta.`


`-[2 cos (frac{2 A+2 C}{2}) cdot cos (frac{2 A-2 C}{2})]`


`-[2 cos ^{2} B+2 cos (A+C) cdot cos (A-C)]`


` [2 cos ^{2} B-2 cos B cos (A-C)]`


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