I have a couple problems I cannot solve, please help me!

1. tan(x)/sec(x) + cot(x)/csc(x) =sin(x)+cos(x)

2 (1-tan(x))^2 = sec^2(x) - 2tan(x)

3. sin(x)/csc(x)-1 + sin(x)/csc(x)+1 = 2tan^2(x)

4. sin(x)[sec(x)-csc(x)] = tan(x)-1

Most of these types of things can be solved if you remember these definitions and identifies:

sin^2(x) + cos^2(x) = 1 (1)

tan(x) = \frac {sin(x)}{cos(x)} (2)

cot(x) = \frac {1}{tan(x)} (3)

sec(x) = \frac {1}{cos(x)} (4)

csc(x) = \frac {1}{sec(x)} (5)

In addition to those, you sometimes need to look up double-angle formulas and a few similar things.

I'll do #1. You try the others.

\frac {tan(x)}{sec(x)} + \frac {cot(x)}{csc(x)} = sin(x) + cos(x)

substituting from #2, #3, #4, and #5:

\LARGE \frac {\left(\frac {sin(x)}{cos(x)}\right)}{\left(\frac {1}{cos(x)}\right)} + \frac {\left( \frac {cos(x)}{sin(x)} \right)}{\left( \frac {1}{sin(x)}\right)}= sin(x) + cos(x)

sin(x) + cos(x) = sin(x) + cos(x)

qed