Question

How do you show that `csctheta = (1+ sec theta)/(tan theta + sintheta)`?

Answer 1

I like to convert everything to sine and cosine and go from there. Use the following identities to do so:

•`csctheta = 1/sintheta`
•`sectheta = 1/costheta`
•`tantheta = sintheta/costheta`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`1/sintheta = (1+ 1/costheta)/(sintheta/costheta + sin theta)`

`1/sintheta = ((costheta + 1)/costheta)/((sin theta + sinthetacostheta)/costheta)`

`1/sintheta = (costheta + 1)/costheta * costheta/(sin theta + sinthetacostheta)`

`1/sintheta = (costheta+ 1)/costheta * costheta/(sintheta(1 + costheta))`

`1/sintheta = 1/sintheta`

`LHS = RHS`

This identity has been proved.

Hopefully this helps!

Answer 2

`RHS= (1+sec theta )/ (tanx + sinx)`

`= (1+sec theta )/ (sinx/cosx + sinx)`

`= (1+sec theta )/ (sinxsecx + sinx)`

`= (1+sec theta )/ (sinx(secx + 1))`

`=1/sinx=cscx=LHS`

Verified