To prove the identity, you'll need these four, more basic identities:
cscx=1/sinx
cotx=cosx/sinx
tanx=sinx/cosx
sin^2x+cos^2x=1
To start the proof, use these identities to write everything in terms of sine and cosine. I'll start with the left-hand side and manipulate it until it looks like the right:
LHS=color(red)(cscx)/(color(blue)(cotx)+color(green)(tanx))
color(white)(LHS)=color(red)(1/sinx)/(color(blue)(cosx/sinx)+color(green)(sinx/cosx))
color(white)(LHS)=color(red)(1/sinxcolor(black)(*sinx))/(color(blue)(cosx/sinxcolor(black)(*sinx))+color(green)(sinx/cosxcolor(black)(*sinx)))
color(white)(LHS)=color(red)(1/color(black)cancelcolor(red)sinx color(black)(*color(red)cancelcolor(black)sinx))/(color(blue)(cosx/color(red)cancelcolor(blue)sinxcolor(black)(*color(red)cancelcolor(black)sinx))+color(green)(sinx/cosxcolor(black)(*sinx)))
color(white)(LHS)=color(red)1/(color(blue)(cosx)+color(green)(sin^2x/cosx))
color(white)(LHS)=(color(red)1*cosx)/(color(blue)(cosx)*cosx+color(green)(sin^2x/cosx)*cosx)
color(white)(LHS)=(color(red)1*cosx)/(color(blue)(cosx)*cosx+color(green)(sin^2x/color(red)cancelcolor(green)cosx)*color(red)cancelcolor(black)cosx)
color(white)(LHS)=color(red)cosx/(color(blue)(cos^2x)+color(green)(sin^2x))
color(white)(LHS)=color(red)cosx/color(darkgreen)1
color(white)(LHS)=color(red)cosx
color(white)(LHS)=RHS
That's the proof. Hope this helped!
