Question

use trig identities to change
\[ \frac{sec^3~\theta}{tan~\theta} \]
to
\[ sec~\theta~tan~\theta~+~csc~\theta \]

Answer 1

$RTP: \frac{sec^{3}(x)}{tan(x)} = sec(x) \cdot tan(x) + cosec(x)$

$LHS = \frac{sec^{3}(x)}{tan(x)}$
$= \frac{sec^{2}(x)}{tan(x)} \cdot sec(x)$

$sec^{2}(x) = tan^{2}(x) + 1$
$LHS = \frac{tan^{2}(x) + 1}{tan(x)} \cdot sec(x)$
$= (tan(x) + \frac{1}{tan(x)}) \cdot sec(x)$
$= (tan(x) + cot(x)) \cdot sec(x)$
$= sec(x) \cdot tan(x) + cot(x) \cdot sec(x)$
$= sec(x) \cdot tan(x) + \frac{1}{cos(x)} \cdot \frac{cos(x)}{sin(x)}$
$= sec(x) \cdot tan(x) + \frac{1}{sin(x)}$

$= sec(x) \cdot tan(x) + cosec(x)$
$= RHS$

$\therefore \frac{sec^{3}(x)}{tan(x)} = sec(x) \cdot tan(x) + cosec(x)$