Question

How do you prove: `(2cos(2x))/(sin(2x))=cot(x)-tan(x)`?

Answer 1

See below for proof.

Explanation:

Things you need to know & remember:

`color(red)(ul("Double angle forumlas""))`
`color(white)("XXX")color(red)(cos(2theta)=cos^2(theta)-sin^2(theta))`
`color(white)("XXX")color(red)(sin(2theta)=2 sin(theta) cos(theta))`

`color(blue)(ul("Relation of " sin " and " cos " to " tan " and "cot))`
`color(white)("XXX")color(blue)(tan(theta)=(sin(theta))/(cos(theta))color(white)("XXX")cot(theta)=cos(theta)/sin(theta)`

`LS = (2cos(2x))/(sin(2x)`

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

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

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

`color(white)("XXX")=cot(x) - tan(x)`

`color(white)("XXX")=RS`