Question

How so you prove: `(sec^2(x)cot(x)-cot(x))/(tan(x)sin(x)+cos(x))=sin(x)`?

Answer 1

Simplify the numerator and denominator separately then do the division

Explanation:

Numerator
`color(red)(sec^2(x)cot(x)-cot(x))`

`color(white)("XXX")color(red)(=cot(x)(sec^2(x)-1))`

`color(white)("XXX")color(red)(=cot(x)(1/cos^2(x)-1))`

`color(white)("XXX")color(red)(=cot(x)((1-cos^2(x))/cos^2(x)))`

`color(white)("XXX")color(red)(=cot(x)((sin^2(x))/(cos^2(x))))`

`color(white)("XXX")color(red)(=cos(x)/sin(x)*((sin^2(x))/(cos^2(x))))`

`color(white)("XXX")color(red)(=sin(x)/cos(x))`

Denominator
`color(blue)(tan(x)sin(x)+cos(x)`

`color(white)("XXX"color(blue)(=(sin(x)/cos(x))*sin(x)+cos(x)`

`color(white)("XXX"color(blue)()=sin^2(x)/cos(x) +cos(x)`

`color(white)("XXX")color(blue)(=(sin^2(x)+cos^2(x))/cos(x)`

`color(white)("XXX")color(blue)(=1/cos(x)`

Resultant Division
`color(red)((sec^2(x)cot(x)-cot(x)))/color(blue)((tan(x)sin(x)+cos(x)))`

`color(white)("XXX")=color(red)((sin(x)/(cos(x)))/color(blue)((1/cos(x)))`

`color(white)("XXX")=sin(x)/cos(x)*cos(x)/1`

`color(white)("XXX")=sin(x)`

QED