Question

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

Answer 1

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(a-b) / (sin a * cos b ) = 1 - cot a * tan b`

To prove this, let's use the following:

1. `cot x = cos x / sin x`

2. `tan x = sin x / cos x`

3. `sin (x - y ) = sin x cos y - cos x sin y`

Now you can prove the identity as follows:

`sin(a-b) / (sin a * cos b ) = (sin a cos b - cos a sin b) / (sin a cos b)`

`color(white)(xxxxxxxx) = (sin a cos b) / (sin a cos b) - (cos a sin b) / (sin a cos b)`

`color(white)(xxxxxxxx) = 1 - (cos a * sin b) / (sin a * cos b)`

`color(white)(xxxxxxxx) = 1 - cos a / sin a * sin b / cos b`

`color(white)(xxxxxxxx) = 1 - cot a * tan b`

Hope that this helped.