F.4 trigonometric functions

(1a) (cos9x/cos3x) + (sin9x/sin3x)
= (cos9x sin3x + sin9x cos3x)/(sin3x cos3x)
= sin(9x+ 3x)/[(1/2)sin6x]
= 2sin12x/sin6x
= 4 sin6x cos6x / sin6x
= 4cos6x
(1b) (cos 9x/cos 3x)+(sin 9x/sin 3x)=2√3
4cos6x = 2√3
cos6x = (√3)/2 = cos(30)
6x = 360n +/- 30
x = 60n +/- 5 where n is an integer
(2) (1 + cosx + sinx)/(1 + cosx - sinx)
= (1 + cosx + sinx)^2/[(1 + cosx - sinx)(1 + cosx + sinx)]
= (1 + cos^2x + sin^2x + 2cosx + 2sinx + 2sinx cosx)/[(1 + cosx)^2 - sin^2x]
= (2 + 2cosx + 2sinx + 2sinx cosx)/(1 + 2cosx + cos^2x - sin^2x)
= (2 + 2cosx + 2sinx + 2sinx cosx)/(2cosx + 2cos^2x)
= (1 + cosx + sinx + sinx cosx)/(cosx + cos^2x)
= (1 + sinx)(1 + cosx)/[cosx(1 + cosx)]
= (1 + sinx)/cosx [Not (1 - sinx)/cosx]