1. 1-sin x + 1-cos x = sin x + cos x-1
cos x sin x sin x cos x

2. sec^2x - tan^2 x + tan x = sin x + cos x
sec x

3. sin^3 x - sin x + cos x = cot x - cos^2 x
sin x

4. 1 + cot A + sec A _ tan A = sin A + cos A
csc A

2. Originally Posted by john edgar
1. 1-sin x + 1-cos x = sin x + cos x-1
cos x sin x sin x cos x
Hint: Multiply both sides of the equation by sin(x) cos(x) and then try it.

Originally Posted by john edgar
2. sec^2x - tan^2 x + tan x = sin x + cos x
sec x
Convert everything to sines and cosines:
$\frac{sec^2(x) - tan^2(x) + tan(x)}{sec(x)} = \frac{\frac{1}{cos^2(x)} - \frac{sin^2(x)}{cos^2(x)} + \frac{sin(x)}{cos(x)}}{\frac{1}{cos(x)}}$

$= \frac{1}{cos(x)} - \frac{sin^2(x)}{cos(x)} + sin(x)$

Originally Posted by john edgar
3. sin^3 x - sin x + cos x = cot x - cos^2 x
sin x
What's $sin^2(x) - 1$? Hint: It's a familiar identity that has been slightly rewritten.

Originally Posted by john edgar
4. 1 + cot A + sec A _ tan A = sin A + cos A
csc A
Convert everything to sines and cosines and simplify:
$\frac{1 + cot(A) + sec(A)}{csc(A)} - tan(A) = \frac{1 + \frac{cos(A)}{sin(A)} + \frac{1}{cos(A)}}{\frac{1}{sin(A)}} - \frac{sin(A)}{cos(A)}$

$= sin(A) + cos(A) + \frac{sin(A)}{cos(A)} - \frac{sin(A)}{cos(A)}$

-Dan