(cos6x.cos3x-sin8x.cosx)/(sin3xsin4x-cos2x.cosx)=tan2x

Question

prove that (cos6x.cos3x-sin8x.cosx)/(sin3xsin4x-cos2x.cosx)=tan2x

Answer 1

sin(A+B)=sin(A)cos(B)+cos(A)sin(B); sin(A-B)=sin(A)cos(B)-cos(A)sin(B); 

sin(A+B)+sin(A-B)=2sin(A)cos(B); sin(A+B)-sin(A-B)=2cos(A)sin(B).

Let A=8x, B=x, then sin(9x)+sin(7x)=2sin(8x)cos(x); and sin(9x)-sin(7x)=2cos(8x)sin(x);

Let A=6x, B=3x, then sin(9x)+sin(3x)=2sin(6x)cos(3x); and sin(9x)-sin(3x)=2cos(6x)sin(3x).

Therefore: sin(9x)+sin(7x)-(sin(9x)+sin(3x))=2sin(8x)cos(x)-2sin(6x)cos(3x).

sin(7x)-sin(3x)=2sin(8x)cos(x)-2sin(6x)cos(3x).

cos(A+B)=cos(A)cos(B)-sin(A)sin(B); cos(A-B)=cos(A)cos(B)+sin(A)sin(B);

cos(A+B)+cos(A-B)=2cos(A)cos(B); cos(A-B)-cos(A+B)=2sin(A)sin(B).

Let A=2x, B=x, then cos(3x)+cos(x)=2cos(2x)cos(x).

Let A=4x, B=3x, then cos(x)-cos(7x)=2sin(4x)sin(3x).

Therefore: cos(3x)+cos(7x)=2cos(2x)cos(x)-2sin(4x)sin(3x).

sin(7x)-sin(3x)=2cos(5x)sin(2x); cos(3x)+cos(7x)=2cos(5x)cos(2x).

(sin(7x)-sin(3x))/(cos(3x)+cos(7x))=2cos(5x)sin(2x)/(2cos(5x)cos(2x))=sin(2x)/cos(2x)=tan(2x) QED