Question

simplify the trig expression....csc^3x-csc^2x-cscx+1

Answer 1

i dont know how to factor it

Answer 2

are you able to factor something like z^3 - z^2 - z + 1 ?

Answer 3

plug cosx = 1

Answer 4

i think i can factor it.

Answer 5

of course now you can. we found that (cosx-1) is a factor

Answer 6

where is cosx-1 coming from

Answer 7

find A,B and C. then you can refactor it

Answer 8

@lochana those are cosecants, not cosines

Answer 9

yes. that's right. I didn't see it clearly. please change cosx into cscx. sorry:(

Answer 10

so plug cscx = 1
now you can say \( csc^3x - csc^2x - cscx + 1 = (cscx -1)(Acsc^2x + Bcscx + C )\)

Answer 11

why am i plugging cscx=1??

Answer 12

because your equation becomes 0 when cscx = 1.

Answer 13

0 means it's a factor. only factors can do that.

Answer 14

im confused

Answer 15

familar with \(x^2 + 4x + 4\)?

Answer 16

no

Answer 17

\(x^2+4x+4 = (x+2)^2\) right?

Answer 18

yes

Answer 19

now plug x = -2 in \((x+2)^2\)

Answer 20

what happened?

Answer 21

it became 0. yes? therefore when you plug x = -2 in \(x^2 + 4x +4\), it becomes 0 since \((x+2)^2 = x^2+4x+4\).

same way, \(csc^3x - csc^2x - cscx +1 \) becomes 0 when \(cscx = 1\) means (cscx-1) is a factor

Answer 22

and A = 1; B = 0; C = -1

Answer 23

so now it is\((cscx-1)(csc^2x - 1 )\), well now it is not that hard. answer will be \((cscx-1)(cscx+1)(cscx -1 )\)