cos2a+cos2b+cos2(a-b)+1
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Let (a+b) = x
[cos(a+b)]^2 = [(1 + cos 2(a+b))/2] (1)
[cos(a-b)]^2 = [(1 + cos 2(a-b))/2] (2)
We'll add (1) + (2):
[(1 + cos 2(a+b) + 1 + cos 2(a-b)/2] = 1 + cos 2a*cos 2b
2/2 + [cos 2(a+b)]/2 + [cos 2(a-b)]/2 = 1 + cos 2a*cos 2b
We'll eliminate 1 both sides:
[cos 2(a+b)]/2 + [cos 2(a-b)]/2 = cos 2a*cos 2b
cos 2(a+b) + cos 2(a-b) = 2cos 2a*cos 2b
cos 2(a+b) = 2[cos (a+b)]^2 - 1
cos 2(a-b) = 2[cos (a-b)]^2 - 1
cos 2a = 2 (cos a)^2 - 1
cos 2b = 2 (cos b)^2 - 1
2cos 2a*cos 2b = 2[2 (cos a)^2 - 1]*[2 (cos b)^2 - 1]
2cos 2a*cos 2b = 2{4(cos a)^2*(cos b)^2-2[(cos a)^2+(cos b)^2] + 1}
[cos (a+b)]^2 = [cos (a+b)][cos (a+b)] = (cosa*cosb-sina*sinb)^2
(cosa*cosb-sina*sinb)^2 = (cosa*cosb)^2 - 2cosa*cosb*sina*sinb + (sina*sinb)^2
2(cosa*cosb-sina*sinb)^2 = 2(cosa*cosb)^2 - 4cosa*cosb*sina*sinb + 2(sina*sinb)^2
2[cos (a+b)]^2 - 1 = 2(cosa*cosb)^2 - 4cosa*cosb*sina*sinb + 2(sina*sinb)^2 - 1 (3)
2[cos (a-b)]^2 - 1 =2(cosa*cosb)^2 + 4cosa*cosb*sina*sinb + 2(sina*sinb)^2 - 1 (4)
We'll add (3) + (4):
2(cosa*cosb)^2 - 4cosa*cosb*sina*sinb + 2(sina*sinb)^2 - 1 +2(cosa*cosb)^2 + 4cosa*cosb*sina*sinb + 2(sina*sinb)^2 - 1
We'll eliminate like terms:
4(cosa*cosb)^2 + 4(sina*sinb)^2 - 2 = 2{4(cos a)^2*(cos b)^2-2[(cos a)^2+(cos b)^2] + 1}.
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