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# prove that: (cos7x-cos8x)/(1+2cos5x)=cos2x-cos3x

Shubham Kamleshbhai Patel
59 Points
6 years ago
L.H.S = (cos7x – cos8x) / 1 + 2cos5x
(Adding cos2x and -cos2x in the numerator)
= (cos7x + cos2x) – (cos8x + cos2x) / 1 + 2cos5x
(Using cosA + cosB = 2cos(A+B/2)cos(A-B/2) )
= (cos7x + cos 2x) – (2cos5x cos3x) / 1 + 2cos5x
(Adding cos3x and -cos3x in the numerator)
= (cos7x + cos2x + cos 3x) – (2cos5x cos3x + cos3x) / 1 + 2cos5x
= (2cos5x cos2x + cos2x) – cos3x (2cos5x + 1) / 2cos 5x + 1
= cos2x (2cos5x + 1) – cos3x (2cos5x + 1) / 2cos5x + 1
= (2cos5x + 1) (cos2x – cos3x) / ( 2cos5x + 1)
= cos2x – cos3x = R.H.S
22 Points
4 years ago
L.H.S = (cos7x – cos8x) / 1 + 2cos5x (Adding cos2x and -cos2x in the numerator) = (cos7x + cos2x) – (cos8x + cos2x) / 1 + 2cos5x (Using cosA + cosB = 2cos(A+B/2)cos(A-B/2) ) = (cos7x + cos 2x) – (2cos5x cos3x) / 1 + 2cos5x (Adding cos3x and -cos3x in the numerator) = (cos7x + cos2x + cos 3x) – (2cos5x cos3x + cos3x) / 1 + 2cos5x = (2cos5x cos2x + cos2x) – cos3x (2cos5x + 1) / 2cos 5x + 1 = cos2x (2cos5x + 1) – cos3x (2cos5x + 1) / 2cos5x + 1 = (2cos5x + 1) (cos2x – cos3x) / ( 2cos5x + 1) = cos2x – cos3x = R.H.SIntegrate it and get answer as[sin2x]/2 - [sin3x]/3Derivative of it would be 3sin3x - 2sin2x