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

1 Answer
May 14, 2016

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) }