How do you factor the expression and use the fundamental identities to simplify ${csc}^{3} x - {csc}^{2} x - csc x + 1$ $csc^3x-csc^2x-cscx+1$ ?

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Answer 1 · 2017-03-28T17:26:11

$x = \pm \frac{π}{2} + 2 k π$ $x=pm pi/2+2kpi$ with $k in ZZ$

Considering $y = cscx$

$y^3-y^2-y+1=0$ has a root for $y = 1$

and

$y^3-y^2-y+1=(y^2-1)(y-1)$

or

$y^3-y^2-y+1=(y+1)(y-1)^2$ so the solutions are

$cscx=pm1$ or

$sinx=pm1$

or

$x=pm pi/2+2kpi$ with $k in ZZ$

How do you factor the expression and use the fundamental identities to simplify csc^3x-csc^2x-cscx+1csc3x−csc2x−cscx+1csc^3x-csc^2x-cscx+1?