#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^@#