How do you factor and simplify $sec x csc x - {csc}^{2} x$ $secxcscx-csc^2x$ ?

Question

2772 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-22T02:28:29

Factor out a $csc x$ $cscx$ :

$\Rightarrow csc x (sec x - csc x)$ $=>cscx(secx - cscx)$

Simplify using the reciprocal identities $csc θ = \frac{1}{sin θ}$ $csctheta = 1/sintheta$ and $sec θ = \frac{1}{cos θ}$ $sectheta = 1/costheta$ .

$\Rightarrow \frac{1}{sin x} (\frac{1}{cos x} - \frac{1}{sin x})$ $=>1/sinx(1/cosx - 1/sinx)$

$\Rightarrow \frac{1}{sin x cos x} - \frac{1}{{sin}^{2} x}$ $=> 1/(sinxcosx) - 1/(sin^2x)$

$\Rightarrow \frac{sin x}{{sin}^{2} x cos x} - \frac{cos x}{{sin}^{2} x cos x}$ $=> sinx/(sin^2xcosx) - cosx/(sin^2xcosx)$

$\Rightarrow \frac{sin x - cos x}{{sin}^{2} x cos x}$ $=>(sinx - cosx)/(sin^2xcosx)$

$\Rightarrow {csc}^{2} x sec x (sin x - cos x)$ $=>csc^2xsecx(sinx - cosx)$

This is as far as we can go. It should be noted that $x \neq 2 π n, \frac{π}{2} + 2 π n$ $x !=2pin, pi/2 + 2pin$

Hopefully this helps!

How do you factor and simplify secxcscx-csc^2xsecxcscx−csc2xsecxcscx-csc^2x?