# How do you simplify sin(a - b)/sina x cosb = 1 - cota x tanb?

Nov 27, 2015

I'm taking a wild guess here that with $x$, you actually meant "x", the multiplication sign.

I'm also taking a second wild guess that "$\cos b$" on your left side should have been a part of the denominator...

So, I think what you would like to prove is

$\sin \frac{a - b}{\sin a \cdot \cos b} = 1 - \cot a \cdot \tan b$

To prove this, let's use the following:

1. $\cot x = \cos \frac{x}{\sin} x$

2. $\tan x = \sin \frac{x}{\cos} x$

3. $\sin \left(x - y\right) = \sin x \cos y - \cos x \sin y$

Now you can prove the identity as follows:

$\sin \frac{a - b}{\sin a \cdot \cos b} = \frac{\sin a \cos b - \cos a \sin b}{\sin a \cos b}$

$\textcolor{w h i t e}{\times \times \times \times} = \frac{\sin a \cos b}{\sin a \cos b} - \frac{\cos a \sin b}{\sin a \cos b}$

$\textcolor{w h i t e}{\times \times \times \times} = 1 - \frac{\cos a \cdot \sin b}{\sin a \cdot \cos b}$

$\textcolor{w h i t e}{\times \times \times \times} = 1 - \cos \frac{a}{\sin} a \cdot \sin \frac{b}{\cos} b$

$\textcolor{w h i t e}{\times \times \times \times} = 1 - \cot a \cdot \tan b$

Hope that this helped.