It has been a really long time since i have done anything Trig and i am having difficulty remembering. i am asked to use the sum to product identities to find an identity for cos x - cos 7 x............can someone plz help me with this?
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cos a - cos b = -2 sin((a+b)/2) sin((a-b)/2)
put a = x and b = 7x and simplify.
put a = x and b = 7x and simplify.
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.. cos(x) - cos(7x)
= cos(x) - [ 64cos^7(x) - 112cos^5(x) + 56cos^3(x) - 7cos(x) ]
= - 64cos^7(x) + 112cos^5(x) - 56cos^3(x) + 8cos(x)
= - cos(x) [ 64cos^6(x) - 112cos^4(x) + 56cos^2(x) - 8 ]
= - 8 cos(x) (cos(x) - 1) (cos(x) + 1) (2cos(x) - 1) (2 cos(x) + 1) (2 cos²(x) - 1)
= cos(x) - [ 64cos^7(x) - 112cos^5(x) + 56cos^3(x) - 7cos(x) ]
= - 64cos^7(x) + 112cos^5(x) - 56cos^3(x) + 8cos(x)
= - cos(x) [ 64cos^6(x) - 112cos^4(x) + 56cos^2(x) - 8 ]
= - 8 cos(x) (cos(x) - 1) (cos(x) + 1) (2cos(x) - 1) (2 cos(x) + 1) (2 cos²(x) - 1)