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tpolwort · Dec 1, 2009, 12:17 PM

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

Perito · Dec 1, 2009, 01:27 PM

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:

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

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

qed