#2sec xsin x + 2 = 4sin x + sec x#
#(2sin x)/(cos x) + 2 = (4sinxcos x + 1)/(cos x)#
#2(sin x + cos x) = 4sinxcos x + 1# (1)
- Multiplying both side by cos x (condition cos x diff. to zero)
Call #(sin x + cos x ) = u#
#u^2 = (sin x + cos x)^2 = sin^2 x + cos^2 x + 2sinxcos x #
#= 1 + 2sin xcos x#.
#2sin xcos x = u^2 - 1#
#4sin xcos x = 2u^2 - 2#
Substitute these values into (1):
#2u = 2u^2 - 2 + 1#
#2u^2 - 2u - 1 = 0#
Solve this quadratic equation for #u = (sin x +cos x)#
#D = d^2 = b^2 - 4ac = 4 + 8 = 12# --> #d = +- 2sqrt3#
There are 2 real roots:
#sin x + cos x = u = 2/2 +- (2sqrt3)/4 = 2 +- sqrt3/2#
#u_1 = 1 + sqrt3/2# (Rejected as #>sqrt2#)
#u_2 = 1 - sqrt3/2 = 0.134#
#sin x + cos x = sqrt2cos (x - pi/4) = 0.134#
#cos (x - pi/4) = 0.134/sqrt2 = 0.0947#
#x - pi/4 = +- 84.564^@#
#x = 84.564 + 45 = 129.564^@# and
#x = 360- 84.564 + 45 = 321.564^@#
