Question

How do you prove `(sinx+cosx)/sinx - (cosx-sinx)/cosx = sec x csc x`?

Answer 1

Please see below.

Explanation:

We know that,

`color(red)((1)cos^2x+sin^2x=1`

`color(blue)((2)1/cosx=secx and 1/sinx=cscx`

Here,

`(sinx+cosx)/sinx-(cosx-sinx)/cosx=secxcscx`

We take,

`LHS=(sinx+cosx)/sinx-(cosx-sinx)/cosx`

`=(cosx(sinx+cosx)-sinx(cosx-sinx))/(sinxcosx)`

`=(cancel(sinxcosx)+cos^2x-cancel(sinxcosx)+sin^2x)/(sinxcosx`

`=color(red)((cos^2x+sin^2x))/(sinxcosx)...tocolor(red)(Apply(1)`

`=color(red)(1)/(sinxcosx)`

`=color(blue)(1/cosx*1/sinx...tocolor(blue)(Apply(2)`

`=color(blue)(secxcscx`

`=RHS`