Hi,
I'm looking for help with two trig proof/verify problems. They are:
1. SINX + COSX . COTX = CSCX
2. ((1 + COSX) / SINX) + (SINX/ (1 + COSX)) = 2CSCX
Thanks in Advance,
Mike Clemmons
1. $\displaystyle \sin{x} + \cos{x}\cot{x} = \sin{x} + \cos{x}\left(\frac{\cos{x}}{\sin{x}}\right)$
$\displaystyle = \frac{\sin^2{x}}{\sin{x}} + \frac{\cos^2{x}}{\sin{x}}$
$\displaystyle = \frac{\sin^2{x} + \cos^2{x}}{\sin{x}}$
$\displaystyle = \frac{1}{\sin{x}}$
$\displaystyle = \csc{x}$.
2. $\displaystyle \frac{1 + \cos{x}}{\sin{x}} + \frac{\sin{x}}{1 + \cos{x}}$
$\displaystyle = \frac{(1 + \cos{x})(1 + \cos{x})}{\sin{x}(1 + \cos{x})} + \frac{\sin{x}\sin{x}}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{(1 + \cos{x})^2 + \sin^2{x}}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{1 + 2\cos{x} + \cos^2{x} + \sin^2{x}}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{1 + 2\cos{x} + 1}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{2 + 2\cos{x}}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{2(1 + \cos{x})}{\sin{x}(1 + \cos{x})}$
$\displaystyle = \frac{2}{\sin{x}}$
$\displaystyle = 2\csc{x}$.