Question

Differentiate `cos(x-2)/sinx`?

Answer 1

`f(x)=cos(x-2)/sinx=(cosxcos2-sinxsin2)/sinx`
`f(x)=cos2cotx-sin2`
`:.f'(x)=cos2(-csc^2x)-0=-cos2csc^2x=-cos2/sin^2x`
OR
Please see below.

Explanation:

We take,

`f(x)=(cos(x-2))/sinx`

Diff.w.r.t. `x`,`"using "color(blue)"Quotient Rule :"`

`color(blue)(d/(dx)(u*v)=(v*(du)/(dx)-u*(dv)/(dx))/v^2`

`f'(x)=(sinx*d/(dx)(cos(x-2))-cos(x-2)d/(dx)(sinx))/((sinx)^2)`

`f'(x)=(sinx(-sin(x-2))-cos(x-2)cosx)/sin^2x`

`f'(x)=-([cos(x-2)cosx+sin(x-2)sinx])/sin^2x`

`f'(x)=-[cos((x-2)-x)]/sin^2x=-cos(-2)/sin^2x`

`f'(x)=-cos2/sin^2x`