# How do you express cot^3theta-cos^2theta-tan^2theta  in terms of non-exponential trigonometric functions?

Jun 9, 2018

$\frac{{\cos}^{5} \left(x\right) - {\sin}^{5} \left(x\right) - {\cos}^{4} \left(x\right) {\sin}^{3} \left(x\right)}{{\sin}^{3} \left(x\right) \cdot {\cos}^{2} \left(x\right)}$

#### Explanation:

We write
$\frac{1}{\tan} ^ 3 \left(x\right) - {\tan}^{2} \left(x\right) - {\cos}^{2} \left(x\right)$

$\frac{1 - {\tan}^{5} \left(x\right)}{\tan} ^ 3 \left(x\right) - {\cos}^{2} \left(x\right)$

$\frac{1 - {\tan}^{5} \left(x\right) - {\tan}^{3} \left(x\right) \cdot {\cos}^{2} \left(x\right)}{\tan} ^ 3 \left(x\right)$

$\frac{1 - {\tan}^{5} \left(x\right) - {\sin}^{2} \frac{x}{\cos} \left(x\right)}{\tan} ^ 3 \left(x\right)$

$\frac{\cos \left(x\right) - \cos \left(x\right) {\tan}^{5} \left(x\right) - {\sin}^{3} \left(x\right)}{\cos \left(x\right) {\tan}^{3} \left(x\right)}$

$\frac{{\cos}^{5} \left(x\right) - {\sin}^{5} \left(x\right) - {\cos}^{4} \left(x\right) {\sin}^{3} \left(x\right)}{{\sin}^{3} \left(x\right) {\cos}^{2} \left(x\right)}$

Aug 10, 2018

$\frac{\cos 3 \theta + 3 \sin \theta}{3 \sin \theta - \sin 3 \theta}$
$- \frac{1}{2} \left(1 + \cos 2 \theta\right) - \frac{1 - \cos 2 \theta}{1 + \cos 2 \theta}$

#### Explanation:

Use

$\cos 3 A = 4 {\cos}^{3} A - 3 \cos A \mathmr{and} \sin 3 A = 3 \sin A - 4 {\sin}^{3} A .$

${\cot}^{3} \theta - {\cos}^{2} \theta - {\tan}^{2} \theta$

$= {\cos}^{3} \frac{\theta}{\sin} ^ 3 \theta - {\cos}^{2} \theta - {\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta$

$= \frac{\cos 3 \theta + 3 \sin \theta}{3 \sin \theta - \sin 3 \theta}$

$- \frac{1}{2} \left(1 + \cos 2 \theta\right) - \frac{1 - \cos 2 \theta}{1 + \cos 2 \theta}$