I want to go from $\displaystyle ac/(a^4sin^4(x)-c^2a^2sin^2(x))^{0.5} $ to $\displaystyle cosec^2(x)/((a/c)^2-1-cot^2(x))^{0.5}$
$\displaystyle \frac{ac}{\sqrt{a^4 \sin ^4 (x) - c^2 a^2 \sin ^2 (x)}} $
to
$\displaystyle \frac{csc ^2 (x)}{\sqrt{\frac{a^2}{c^2} -1 - \cot ^2 (x)}} $
you can divide the denominator and nominator by ac note $\displaystyle ac = \sqrt{a^2c^2} $
take $\displaystyle \sin ^4 (x) $ at the square root denominator as a factor
We are given:
$\displaystyle \frac{ac}{\sqrt{a^4\sin^4(x)-c^2a^2\sin^2(x)}}$
Rewriting as:
$\displaystyle \frac{ac}{\sqrt{a^2c^2\sin^4(x)\left[\frac{a^2}{c^2}-\frac{1}{\sin^2(x)}\right]}}$
$\displaystyle =\frac{ac}{ac\sin^2(x)\sqrt{\left(\frac{a}{c} \right)^2-\frac{\sin^2(x)+\cos^2(x)}{\sin^2(x)}}}$
$\displaystyle =\frac{1}{\sin^2(x)\sqrt{\left(\frac{a}{c}\right)^ 2-\frac{\sin^2(x)}{\sin^2(x)}-\frac{\cos^2(x)}{\sin^2(x)}}}$
$\displaystyle =\frac{\csc^2(x)}{\sqrt{\left(\frac{a}{c}\right)^2-1-\cot^2(x)}}$