1. Help proving

I am having problems sproving a couple of equations. Thank you in advance
1.)
$
\tan{x}+cot{x}= \frac{csc^2}{cot{x}}
$

2.) 1-cosx
sinx
= sinx
1+cosx

2. Originally Posted by nightrider456
I am having problems sproving a couple of equations. Thank you in advance
1.)
$
\tan{x}+cot{x}= \frac{csc^2}{cot{x}}
$

2.) 1-cosx
sinx
= sinx
1+cosx
2. $\frac{1 - \cos{x}}{\sin{x}} = \frac{1 - \cos{x}}{\sin{x}}\cdot\frac{1 + \cos{x}}{1 + \cos{x}}$

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

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

$= \frac{\sin{x}}{1 + \cos{x}}$.

3. Originally Posted by nightrider456
I am having problems sproving a couple of equations. Thank you in advance
1.)
$
\tan{x}+cot{x}= \frac{csc^2}{cot{x}}
$

2.) 1-cosx
sinx
= sinx
1+cosx
1) Recall that $\sin^2{x} + \cos^2{x} = 1$.

Dividing both sides by $\sin^2{x}$ gives

$1 + \cot^2{x} = \csc^2{x}$.

So $\frac{\csc^2{x}}{\cot{x}} = \frac{1 + \cot^2{x}}{\cot{x}}$

$= \frac{1}{\cot{x}} + \cot{x}$

$= \tan{x} + \cot{x}$.