Question

Can anyone help prove this trigonometric identity?

`sinx/(1+cos x) + cos x/sin x = csc x`

Answer 1

Cross multiply and simplify LHS (as below)

Explanation:

To prove: `sinx/(1+cosx) + cosx/sinx = cscx`

`LHS = sinx/(1+cosx) + cosx/sinx`

`=(sin^2x+cosx+cos^2x)/(sinx+sinxcosx)`

Since: `sin^2x + cos^2x=1`

`LHS= (1+cosx)/(sinx(1+cosx))`

`= cancel(1+cosx)*1/(sinx*cancel(1+cosx))`

`=1/sinx = cscx = RHS`

Answer 2

`sinx/(1+cosx)+cosx/sinx = ((sinx))/((1+cosx)) * ((1-cosx))/((1-cosx))+cosx/sinx`

`= (sinx(1-cosx))/(1-cos^2x) +cosx/sinx`

`= (sinx(1-cosx))/(sin^2x) +cosx/sinx`

`= (1-cosx)/sinx +cosx/sinx`

`= (1-cosx+cosx)/sinx`

`= 1/sinx`

`= cscx`