Establish the identity.
[sin^3 (x) + cos^3 (x)]/[1 - 2 cos^2 (x)] =
[sec(x) - sin(x)]/[tan(x) - 1]
Consider the LHS:
$\displaystyle \frac {\sin^3 x + \cos^3 x}{1 - 2 \cos^2 x} = \frac {(\sin x + \cos x)(\sin^2 x - \sin x \cos x + \cos^2 x)}{- \cos 2x}$
$\displaystyle = \frac {(\sin x + \cos x)(1 - \sin x \cos x)}{- \cos 2x}$
Now consider the RHS:
$\displaystyle \frac {\sec x - \sin x}{\tan x - 1} = \frac {\frac 1{\cos x} - \sin x}{\frac {\sin x}{\cos x} - 1}$
$\displaystyle = \frac {\frac {1 - \sin x \cos x}{\cos x}}{ \frac {\sin x - \cos x}{\cos x}}$
$\displaystyle = \frac {1 - \sin x \cos x}{\sin x - \cos x}$
$\displaystyle = \frac {1 - \sin x \cos x}{\sin x - \cos x} \cdot \frac {\sin x + \cos x}{\sin x + \cos x}$
$\displaystyle = \frac {(\sin x + \cos x)(1 - \sin x \cos x)}{- \cos 2x}$
$\displaystyle = LHS$
Thus, $\displaystyle LHS \equiv RHS$
Hello, magentarita!
The left side has a sum of cubes ... factor it.Establish the identity: .$\displaystyle \frac{\sin^3\!x + \cos^3\!x}{1 - 2\cos^2\!x}\;= \;\frac{\sec x - \sin x}{\tan x - 1}$
. . . $\displaystyle \frac{(\sin x + \cos x)(\sin^2\!x - \sin x\cos x + \cos^2\!x)}{1 - 2 \cos^2\!x}$
. . $\displaystyle = \;\frac{(\sin x + \cos x)(\overbrace{\sin^2\!x + \cos^2\!x}^{\text{This is 1}} - \sin x\cos x)}{\underbrace{(1 - \cos^2\!x)}_{\text{This is }\sin^2\!x} - \cos^2\!x}$
. . $\displaystyle = \;\frac{(\sin x + \cos x)(1 - \sin x\cos x)}{\sin^2\!x - \cos^2\!x}$
. . $\displaystyle = \;\frac{{\color{red}\rlap{////////////}}(\sin x + \cos x)(1 - \sin x\cos x)}{{\color{red}\rlap{////////////}}(\sin x + \cos x)(\sin x - \cos x)} $
. . $\displaystyle = \;\frac{1 - \sin x\cos x}{\sin x - \cos x}$
Divide top and bottom by $\displaystyle \cos x$
. . $\displaystyle \frac{\dfrac{1}{\cos x} - \dfrac{\sin x{\color{red}\rlap{/////}}\cos x}{{\color{red}\rlap{/////}}\cos x}} {\dfrac{\sin x}{\cos x} - \dfrac{{\color{red}\rlap{/////}}\cos x}{{\color{red}\rlap{/////}}\cos x}} $
. . $\displaystyle = \;\frac{\sec x - \sin x}{\tan x - 1}$
skeeter had a valid point as well. the secant is undefined for the specified values of x. so that may cause problems. after graphing the two functions, i saw that the graphs were the same and so i overlooked it. i guess the point here was to do the algebraic manipulation to show you know your trig identities and how to use them