Johnny Neutron said:
sin (X) / Cos (x) - 1 = show work
What?
Sec ^2/ cot (x) - Tan ^3x = Tan X
[tex]\frac{\sec ^2 x}{\cot x} - \tan ^3 x = \tan x[/tex]
Note, [itex]\cot x[/itex] must not be zero. Now, multiply by [itex]\cot x[/itex]:
[tex]\sec ^2 x - \tan ^2 x = 1 \ \dots \ (1)[/tex]
Note, if [itex]\cot x[/itex] were zero, then [itex]\cos x[/itex] would have to be zero (since [itex]\cot x = \frac{\cos x}{\sin x}[/itex]), but since it's not, then [itex]\cos x \neq 0[/itex]. So, we can multiply both sides by [itex]\cos ^2 x[/itex]:
[tex]1 - \sin ^2 x = \cos ^2 x[/tex]
[tex]\sin ^2 x + cos ^2 x = 1[/tex]
This is a basic identity you should know. In fact (1) is a commonly used identity too, but I figured I'd get you down at least this far. I assume you won't have to prove this. If you do, then you know that [itex]\sin x[/itex] is the ratio of the side opposite the angle x in a right triangle to the hypoteneuse. You should also know the definition for [itex]\cos x[/itex]. With these two definitions and the Pythagorean Theorem, you should be able to prove those two identities.
Sec x - Cos x/tanx = sinx
As a general approach to any of these kinds of problems, express everything in terms of sine and cosine. Mutiplying both sides by [itex]\sin x \cos x[/itex]:
[tex]\sin x - \cos ^3 x = \sin ^2 x \cos x[/tex]
[tex]\sin x = \cos x(\cos ^2 x + \sin ^2 x)[/tex]
[tex]\sin x = \cos x[/tex]
This is wrong. Try x = 32 degrees. It doesn't work. I guess it's a trick question or you mistyped (or I made a mistake).