Question

Question #22cd1

Answer 1

`-secx`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`cscx=1/sinx,cotx=cosx/sinx,secx=1/cosx`

`'and "sin(-x)=-sinx`

`"numerator "=csc^2x-cot^2x`

`color(white)("numeratpr ")=1/sin^2x-cos^2x/sin^2x`

`color(white)("numerator ")=(1-cos^2x)/(sin^2x`

`color(white)("numerator ")=sin^2x/sin^2x=1`

`>"denominator "=sin(-x)cotx`

`color(white)("denominator ")=-sinx xxcosx/sinx=-cosx`

`rArr(csc^2x-cot^2x)/((sin)(-x)cotx`

`=1/(-cosx)=-secx`

Answer 2

`-cscxsecx, or, -2csc2x`.

Explanation:

Since, `csc^2x=1+cot^2x, and sin(-x)=-sinx`, we have,

`(csc^2x-cot^2x)/[sin(-x)cosx]=1/(-sinxcosx)=-1/sinx*1/cosx`

`=-cscxsecx, or,`

`=-2/(2sinxcosx)=-2/(sin2x)=-2csc2x`.