Question

How do you simplify tanx+cotx/cotx?

Answer 1

`1/cos^2x`

Explanation:

Identities:

1) `color(red)bb(tanx=sinx/cosx)`

2) `color(red)bb(cotx=cosx/sinx)`

3) `color(red)(bb(sin^2x+cos^2x=1)`

Rewrite:

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

Add fractions in numerator:

`((sin^2x+cos^2x)/(cosxsinx))/(cosx/sinx)`

By identity 3:

`(1/(cosxsinx))/(cosx/sinx)`

Multiply by `sinx`

`(sinx/(cosxsinx))/(sinxcosx/sinx)`

Cancel:

`(cancel(sinx)/(cosxcancel(sinx)))/(cancel(sinx)cosx/cancel(sinx))`

`(1/cosx)/cosx`

Divide by `cosx`

`(1/(cosxcosx))/(cosx/cosx)`

`(1/(cosxcosx))/(cancel(cosx/cosx))`

`1/cos^2x`