cos 4x sin 3x - cos2xsinx /cos4x sinx+cos 6x cosx = tan 2 x
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Answers
Answer:
Step-by-step explanation:
cos 4x sin 3x - cos2xsinx /cos4x sinx+cos 6x cosx
//multiply and divide by 2.
=> 2Cos4xSin3x - 2Cos2xSinx / 2Cos4xSinx + 2Cos6xCosx
We know that 2 cos a sin b = sin (a + b) - sin (a - b)
2 sin a sin b = cos(a - b) - cos(a + b)
2 cos a cos b = cos(a + b) + cos(a - b)
=> sin(4x + 3x) - sin(4x-3x) - sin(2x+x) - sin(2x-x)/cos(4x-x) - cos(4x+x) + cos(6x + x) + cos(6x - x)
=> sin 7x - sin x - sin 3x + sin x / cos 3 x - cos 5 x + cos 7x + cos 5x
=> sin 7 x - sin 3 x / cos 3 x + cos 7 x
We know that sina - sinb = 2cos(a+b/2)sin(a-b/2)
cosa - cosb = 2cos(a+b/2)cos(a-b/2)
=> 2 cos(7x+3x/2) sin(7x-3x/2) /2 cos(3x+7x/2) cos(7x - 3x/2)
=> 2 cos5xsin2x / 2cos5xcos2x
=> sin2x/cos 2 x
=> Tan2x
= R.H.S
Hence proved.