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!
