Question

Factor and simplify using fundamental identities: csc^3 x - csc^2 x – csc x + 1
The textbook answer is: cot^2 x(csc x – 1).
I tried to factor by grouping thus: (csc^3x – csc x) – (csc^2x + 1). But after 1+cot^2 x(cscx – 1) – (csc x +1) I could not make progress.

Answer 1

you did something wrong in your grouping there...

Answer 2

it should be (csc^3 x - csc x) - (csc^2 x - 1)

does that make sense?

Answer 3

because it's like you're factoring out -1 in the second group

Answer 4

so if you factor out csc x in the first group you get \[\implies \csc x(\csc^2 x - 1) - (\csc^2 x - 1)\]
got it now?

Answer 5

...feel free to ask if there is something you dont understand in what i jus said @SamandAlex

Answer 6

yes lgbasallote is writehave a look again:
.........csc^3x - csc^2x - cscx + 1
..........(csc^3x - cscx) + (-csc^2x + 1)
.........cscx(csc^2 - 1) -(csc^2x-1) remember you need same signs inside the brackets
therefore (csc^2x - 1) (cscx - 1)
remember this identity:cscx=1/sinx; therefore csc^2x = 1/sin^2x ; substitute this in the brachet you'll get (1/sin^2 - 1) (cscx -1)... in the first brcket u'll get (1 - sin^2x)/sin^2x : look at this numerator remember again sin^2x + cos^2x = 1 ; therefore cos^2x = 1 - sin^2x : substitute this u'll get ( cos^2x/sin^2x) (cscx - 1) ; cos^2x/sin^2x = cot^2x therefore ur text book was right the answer is cot^2x(cscx - 1)

Answer 7

Thank you. Igbasallote and hubertH. That was very helpful.

Answer 8

you're welcome