Find the value of cos(4x) if 3 / 1-cos(x)-sin(1) = 4.
Answers
cos (4x) = -97/128 if 3/(1 - cosx - sinx) = 4
Given : 3/(1 - cosx - sinx) = 4
To Find : cos (4x)
3/(1 - cosx - sinx) = 4
=> 3 = 4 - 4(cosx + six)
=> -1 = -4(cosx + six)
=> (cosx + sinx) = 1/4
Squaring both sides
cos²x + sin²x + 2cosxsinx = 1/16
Identity cos²x + sin²x = 1 and sin(2x) = 2sinxcosx
=> 1 + sin(2x) = 1/16
=> sin(2x) = -15/16
Identity cos(2x) = 1- 2sin²x
cos(4x) = 1 - 2sin²(2x)
Substitute sin(2x) = -15/16
=> cos(4x) = 1 - 2( -15/16)²
=> cos (4x) = 1 - 2(225/256)
=> cos(4x) = (256 - 450)/256
=> cos(4x) = -194/256
=> cos (4x) = -97/128
=> cos (4x) = -0.7578125
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