cos(a + b + c) + cos(b + c - a) + cos(c + a - b) + cos(a + b - c)
= (cos(a + b + c) + cos(b + c - a)) + (cos(c + a - b) + cos(a + b - c))
= 2 × cos((a + b + c + b + c - a)/2) × cos((a + b + c - b - c + a/2) + 2 × cos((c + a - b + a + b - c)/2) × cos((c + a - b - a - b + c)/2)
= 2 × cos((2b + 2c)/2) × cos(2a/2) + 2 × cos(2a/2) × cos((2c - 2b)/2)
= 2 × cos(b + c) × cosa + 2 × cosa × cos(c - b)
= 2 × cosa × (cos(b + c) + cos(b - c))
= 2 × cosa × ((cosb × cosc - sinb × sinc) + (cosb × cosc + sinb × sinc))
= 2 × cosa × 2(cosb × cosc)
= 4 × cosa × cosb × cosc
GOOD LUCK!