(cscx + secx)/(sinx + cosx) = cotx + tanx, please help prove!

Question

I need help proving that this equal each other...I've tried a ton of different ways and can't figure it out.  Thanks in advance!

Answer 1

Let c=cos(x) and s=sin(x), then cot(x)=c/s and tan(x)=s/c.

So ((1/s)+(1/c))/(s+c)=[(c+s)/(sc)](s+c)=1/sc (left-hand side),

c/s+s/c=(s²+c²)/sc, but s²+c²=1 so c/s+s/c=1/sc (right-hand side).

left-hand side=right-hand side QED