Question

How do you write `csc(2x) / tanx` in terms of sinx?

Answer 1

`1/ {2 sin^2(x) }`

Explanation:

Useful Trig ID's

Definitions of functions
`csc (x) = 1/sin (x)`

`tan (x) = sin (x) / cos (x)`

Sums of Angles Formula
`sin(x+y)=sin(x) cos(y)+cos(x)sin(y)`
Which gives the double well known double angle formula
`sin(2x)=2 sin(x)cos(x)`

We start with our ID, sub in the basic definition and use some fraction rules to get the following.

`csc(2x)/tan(x) ={1/sin(2x)} / {sin (x)/cos(x)} = 1/ sin (2x) cos (x)/sin(x)`

We replace `sin(2x)` with `2 sin(x) cos(x)`

`= 1/ {2 sin(x)cos(x) } cos (x)/sin(x)`

The cosine's cancel
`= 1/ {2 sin(x) } 1/sin(x)`
leaving us with

`= 1/ {2 sin^2(x) }`