# How do you prove (sectheta - tantheta) (csctheta +1)=cottheta?

Jul 5, 2016

The following identities will be necessary for this problem:

$\sec \theta = \frac{1}{\cos} \theta$

$\csc \theta = \frac{1}{\sin} \theta$

$\tan \theta = \sin \frac{\theta}{\cos} \theta$

$\cot \theta = \cos \frac{\theta}{\sin} \theta$

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

Now, we have what we need to prove:

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

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

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

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

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

Identity proved!!

Practice exercises:

Prove the following identities:

a) $\sin \theta + {\cos}^{2} \frac{\theta}{1 + \sin \theta} = 1$

b) $\sin \theta \left(\csc \theta - \sin \theta\right) = {\cos}^{2} \theta$

exercises taken from http://www.swrschools.org/assets/algebra_2_and_trig/chapter12.pdf

Hopefully this helps and good luck!