Question

How do you prove that `Cosx(Tanx+Cotx) = cscx`?

Answer 1

Please see below.

Explanation:

`cosx(tanx+cotx)`

= `cosx(sinx/cosx+cosx/sinx)`

= `cosx((sin^2x+cos^2x)/(sinxcosx))`

= `cosx xx 1/(sinxcosx)`

= `1/sinx`

= `cscx`

Answer 2

Please see below.

Explanation:

We know that,

#color(red)((1)tantheta=sintheta/costheta and cot theta=costheta/sintheta#

`color(blue)((2)sin^2theta+cos^2theta=1`

Given that,

`cosx(tanx+cotx)=cscx`

`LHS=cosx(tanx+cotx)`

`color(white)(LHS)=cosx{sinx/cosx+cosx/sinx}....tocolor(red)(Apply(1)`

`color(white)(LHS)=cosx{(sin^2x+cos^2x)/(cosxsinx)}...color(blue)(Apply(2)`

`color(white)(LHS)=cosx{1/(sinxcosx)}`

`color(white)(LHS)=1/sinx`

`color(white)(LHS)=cscx`

`:.LHS=RHS`