How to verify or prove this identiy?Here is the identity I need to prove: (sin x + cos x)(tan x + cot x) = sec x + csc x Thanks to any answer!

First you need to remember that tan x is the same as sin
x/cos x,     cot x = cos x/sin x, sec x = 1/cos x, and csc x = 1/sin
x.

Then if you look at the left side of the equation and
multiple out the terms using FOIL, you'll get:

sin^2 x/cos
x + cos x + sin x + cos^2 x/sin x = sec x + csc x

Since you
have fractions on the left side, and you are adding, you need common denominators, so I
changed cos x to cos^2 x/cos x and sin x to sin^2 x/sin x.  Then you can add the two
fractions with cos x in the denominator and the two fractions with the sin x in the
denominator, this gives you:

(sin^2 x +cos^2 x)/cos x +
(sin^2 x + cos^2 x)/sin x = sec x + csc x

The last thing
you need to remember is the identity: sin^2 x + cos^2 x = 1, so you can replace both of
the numerators with one and this gives you:

1/cos x + 1/sin
x = sec x + csc x

Now, by the definitions of secant and
cosecant that I listed at the top, that proves the
identity.

Hopefully that helps.