How do you express sin theta - cottheta + tan theta  in terms of cos theta ?

May 29, 2016

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

$= \frac{\sin \theta \times \sin \theta \times \cos \theta}{\sin \theta \times \cos \theta} + \frac{\cos \theta \times \cos \theta}{\cos \theta \sin \theta} + \frac{\sin \theta \times \sin \theta}{\cos \theta \sin \theta}$

$= \frac{{\sin}^{2} \theta \cos \theta + {\cos}^{2} \theta + {\sin}^{2} \theta}{\sin \theta \cos \theta}$

$= \frac{\left(1 - {\cos}^{2} \theta\right) \left(\cos \theta\right)}{\sin \theta \cos \theta}$

$= \frac{1 - {\cos}^{2} \theta}{\sin} \theta$

$= \sqrt{{\left(\frac{1 - {\cos}^{2} \theta}{\sin} \theta\right)}^{2}}$

$= \sqrt{\frac{1 - 2 {\cos}^{2} \theta + {\cos}^{4} \theta}{{\sin}^{2} \theta}}$

$= \sqrt{\frac{1 - 2 {\cos}^{2} \theta + {\cos}^{4} \theta}{1 - {\cos}^{2} \theta}}$

=sqrt((-(cos2theta) + cos^4theta)/(1 - cos^2theta)

Certainly a long proof, but it works, thankfully!

Hopefully this helps, and have a great day!