First, we need to find out whether this is an identity (to prove) or an equation (to solve). Let x=0:
(1-sin(0)tan(0))/(1+sec(0))=½≠sec(0) because sec(0)=1. So it's an equation that has to be solved for x. (1-sin(x)tan(x))/(1+sec(x))-sec(x)=0 has to be solved.
(1-sin2(x)/cos(x))/(1+1/cos(x))=1/cos(x),
(cos(x)-sin2(x))/(cos(x)+1)=1/cos(x),
(cos2(x)+cos(x)-1)/(cos(x)+1)=1/cos(x).
Let y=cos(x):
(y2+y-1)/(y+1)=1/y,
y(y2+y-1)=y+1 (cross-multiplication),
y3+y2-y=y+1,
y3+y2-2y-1=0.
This cubic has three solutions, but only one is within range for cosine.
Let f(y)=y3+y2-2y-1, then f'(y)=3y2+2y-2, and use Newton's Method to find the solution:
yn+1=yn-f(yn)/f'(yn). Start with y=0, then y1=-½, and eventually arrive at y=-0.4450 approx. cos(x)=y so x=116.426° approx.
This is one solution to the question for x between 0 and 360°. The other is 360-116.426=243.574°. However, I think the question has been misstated, because the solution (which does satisfy the original equation) is too complicated.