Prove sinx/secx+tanx-1 + cosx/cosec x+cotx-1 = 1

If the given expression is sinx/(sec x+tan x-1) +
cosx/(cosec x+cot x-1)=1, we'll do the following steps:

sin
x/(1/cos x+  sin x/cos x - 1) + cos x/(1/sin x + cos x/sin x - 1) =
1

We've replaced sec x=1/cos x; tan x = sin x/cos x ; cosec
x = 1/sin x and cot x = cos x/sin x

sin x/(1/cos x+  sin
x/cos x - 1) + cos x/(1/sin x + cos x/sin x - 1) = 1

sin
x*cos x/(1+sin x- cos x) + sin x*cos x/(1 + cos x - sin x) =
1

We'll multiply the 1st fraction by (1 + cos x - sin x)
and the 2nd fraction by (1+sin x- cos x):

sin x*cos x(1 +
cos x - sin x + 1+sin x- cos x)/(1 + cos x - sin x)*(1+sin x- cos x) =
1

We'll eliminate like terms within brackets and we'll
multiply the right side by (1 + cos x - sin x)*(1+sin x- cos
x):

2 sin x*cos x = (1 + cos x - sin x)*(1+sin x- cos
x)

We'll remove the
brackets:

2 sin x*cos x = 1+sin x- cos x + cos x + sin
x*cos x - (cosx)^2 - sin x - (sin x)^2 + sin x*cos x

But
(sin x)^2 + (cosx)^2 = 1

2 sin x*cos x = 1 - 1 + 2sin x*cos
x

2 sin x*cos x =  2sin x*cos
x

Since both sides are equal, then the given
expression represents an identity.