Question

How do you prove that `\frac { \cos 7x + \cos 5x } { \sin 7x - \sin 5x } = \cot x`?

Answer 1

See the proof below

Explanation:

We need

`cosa+cosb=2cos((a+b)/2)cos((a-b)/2)`

`sina-sinb=2cos((a+b)/2)sin((a-b)/2)`

`cotx=cosx/sinx`

Here,

`a=7x` and `b=5x`

Therefore,

`LHS=(cos7x+cos5x)/(sin7x-sin5x)=(2cos((7x+5x)/2)cos((7x-5x)/2))/(2cos((7x+5x)/2)sin((7x-5x)/2))`

`=(cos(6x)cos(x))/(cos(6x)sin(x))`

`=cosx/sinx`

`=cotx`

`=RHS`

`QED`