# How do you verify the identify (sintheta+costheta)/sintheta=1+cottheta?

Aug 2, 2016

Recall that $\tan \theta = \frac{\sin \theta}{\cos} \theta$ and $\cot \theta = \frac{1}{\tan} \theta$

#### Explanation:

$\frac{\sin \theta + \cos \theta}{\sin \theta} = \frac{\sin \theta}{\sin \theta} + \frac{\cos \theta}{\sin \theta} = 1 + \frac{1}{\tan \theta} = 1 + \cot \theta$

Aug 2, 2016

see explanation.

#### Explanation:

There are 3 possible approaches that may be taken.

(1) Manipulate the left side to obtain the right side.

(2) Manipulate the right side to obtain the left side.

(3) Manipulate both sides until they are the same.

Using (1) along with the following $\textcolor{b l u e}{\text{ trig. identity}}$

$\textcolor{\mathmr{and} a n \ge}{\text{ Reminder}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\cot \theta = \frac{\cos \theta}{\sin \theta}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

left side $= \frac{\sin \theta + \cos \theta}{\sin} \theta$

divide terms on numerator by $\sin \theta$

$\Rightarrow \frac{\cancel{\sin \theta}}{\cancel{\sin \theta}} + \frac{\cos \theta}{\sin \theta}$

$\Rightarrow \frac{\sin \theta + \cos \theta}{\sin} \theta = 1 + \cot \theta = \text{ right side"rArr" verified}$