Simplify: (sin(x-pi))/(cos (pi+x)) - (cos (pi/2-x))/(sin(-pi-x))

We have to simplify: (sin(x-pi))/(cos (pi+x)) - (cos
(pi/2-x))/(sin(-pi-x))

We use the relation sin(a+b) = sin
a*cos b + cos a*sin b and cos( a +b) = cos a*cos b - sin a * sin
b.

(sin(x-pi))/(cos (pi+x)) - (cos
(pi/2-x))/(sin(-pi-x))

=>[(sin x * cos pi - cos
x*sin pi) / (cos pi* cos x - sin pi * sin x)] - [(cos pi/2 *cos x + sin pi/2 *sin x) /
(-sin pi*cos x - cos pi*sin x)]

sin pi = 0, cos pi = -1,
sin pi/2 = 1 and cos pi/2 = 0

=> [(- sin x) / (-cos
x)] - [(sin x) / (+sin x)]

=> tan x -
1

The required simplified result is tan x -
1.