Question

Sin^2xsecxcscx=tanx?

Answer 1

To prove this identity, these other identities are needed:

`secx=1/cosx`

`cscx=1/sinx`

`tanx=sinx/cosx`

Here's the proof (I color-coded some parts so it's easier to follow):

`LHS=color(red)(sin^2x)color(blue)secxcolor(green)cscx`

`color(white)(LHS)=color(red)(sin^2x)*color(blue)secx*color(green)cscx`

`color(white)(LHS)=color(red)(sin^2x/1)*color(blue)(1/cosx)*color(green)(1/sinx)`

`color(white)(LHS)=(color(red)(sin^2x)*color(blue)1*color(green)1)/(color(red)1*color(blue)cosx*color(green)sinx)`

`color(white)(LHS)=color(red)(sin^2x)/(color(blue)cosx*color(green)sinx)`

`color(white)(LHS)=color(red)((sinx)^2)/(color(blue)cosx*color(green)sinx)`

`color(white)(LHS)=(color(red)sinx*color(red)sinx)/(color(blue)cosx*color(green)sinx)`

`color(white)(LHS)=(color(red)sinx*cancelcolor(red)sinx)/(color(blue)cosx*cancelcolor(green)sinx)`

`color(white)(LHS)=color(red)sinx/color(blue)cosx`

`color(white)(LHS)=color(purple)tanx`

`color(white)(LHS)=RHS`

That's the proof. Hope this helped!