Prove (1+secx)/(tanx+sinx)=cscx

Answer 1

Let's use a few identities:

secx = 1/cosx
tanx = sinx/cosx
cscx = 1/sinx

Therefore:

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

Change 1 in the numerator to cosx/cosx, which is the equivalent of 1. Also, multiply sinx in the denominator by cosx/cosx.

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

(cosx + 1)/cosx
---------------- = cscx
[sinx + sinx(cosx)]/cosx

(cosx + 1)/[sinx + sinx(cosx)] = cscx

(cosx + 1)/sinx(cosx + 1) = cscx

1/sinx = cscx

cscx = cscx

I hope this helps.