Question

How do you verify `(tan^3(x) - 1) / (tan(x) - 1) = sec^2(x) + tan(x)`?

Answer 1

LHS =
`(tan^3(x)-1)/(tan(x)-1)`
`"Since " a^3-b^3=(a-b)*(a^2+a*b+b^2)`, we may write the LHS
`= (cancel((tan(x)-1))*(tan^2(x)+tan(x)+1))/cancel((tan(x)-1))`

`=cancel(tan^2(x))+tan(x)+Sec^2(x)-cancel(tan^2(x))` `["Since " Sec^2(x)-tan^2(x) = 1]`
`=Sec^2(x)+tan(x) =`RHS Proved.