how do I prove that tan theta/ sin theta= sec theta

Remember that tanθ=sinθ/cosθ

so you have tanθ/sinθ this is the same as (sinθ/cosθ)*(1/sinθ)

(it is easier to see if you wrote this out on paper)

this cancels the sinθ and you end up with 1/cosθ

Now remember that this identity is equal to secθ

(you know? Cosecθ=1/sinθ secθ=1/cosθ... ahh now you remember! ). Hope this helps ;p