can i use factoring to simplify this trig identity?

Question

the problem is sinx + cotx * cosx i know the answer is cscx and i know how to get it but i want to know if i can do factoring to get it bc i tried to but it wont give me the answer .
this is the step i went through:
1) sinx + cotx * cosx turns into sinx +(cosx/sinx)*cosx
2) i try to factor out sinx so that i would get sinx(1+ cosx * cosx)
3) that left me with sinx(1+cos^(2) x) that's where i im lost can anyone enlighten me plz

Answer 1

cot = cos/sin, so you have

sin + cos^2/sin
= (sin^2+cos^2)/sin
= 1/sin
= csc

Answer 2

im well aware of that steve thank you for answering but i really wanted to know was is it at all possible to use factoring to solve this like i have up there

Answer 3

I don't see any way to use factoring. You don't in fact come up with

sin(1+cos^2)

because you have that pesky 1/sin under the cos^2.

If you try to fractor out the sin, you get

sin(1+cos^2/sin^2)

and again you end up with sin^2+cos^2 on top.

Answer 4

We could do some " silly" factoring

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