Tried all I could on this one. I think I'm making it out to be harder than it's supposed to be...
(sec(x))/(sec(x)-tan(x))=sec^2(x)+sec(x)tan(x)
Any help would be greatly appreciated.
Hey, try expressing sec(x) and tan(x) in terms of sines and cosines first, cancel what you can.
You should get $\displaystyle \frac{1}{1 - \sin (x)} $
The trick here is to multiply the numerator and denominator by $\displaystyle 1 + \sin (x) $
Then use the identity $\displaystyle \sin^2 x + \cos^2 x = 1$ on the denominator.
Convert back to sec and tan and that should be it.
$\displaystyle \sec^2{x} + \sec{x}\tan{x} = \sec{x}(\sec{x} + \tan{x})$
$\displaystyle = \frac{\sec{x}(\sec{x} + \tan{x})(\sec{x} - \tan{x})}{\sec{x} - \tan{x}}$
$\displaystyle = \frac{\sec{x}(\sec^2{x} -\tan^2{x}) }{\sec{x} - \tan{x}}$
$\displaystyle = \frac{\sec{x}(1)}{\sec{x} - \tan{x}}$
$\displaystyle = \frac{\sec{x}}{\sec{x} - \tan{x}}$.