Useful Trig ID's
Definitions of functions
csc (x) = 1/sin (x)csc(x)=1sin(x)
tan (x) = sin (x) / cos (x)tan(x)=sin(x)cos(x)
Sums of Angles Formula
sin(x+y)=sin(x) cos(y)+cos(x)sin(y) 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) sin(2x)=2sin(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)csc(2x)tan(x)=1sin(2x)sin(x)cos(x)=1sin(2x)cos(x)sin(x)
We replace sin(2x)sin(2x) with 2 sin(x) cos(x)2sin(x)cos(x)
= 1/ {2 sin(x)cos(x) } cos (x)/sin(x)=12sin(x)cos(x)cos(x)sin(x)
The cosine's cancel
= 1/ {2 sin(x) } 1/sin(x)=12sin(x)1sin(x)
leaving us with
= 1/ {2 sin^2(x) } =12sin2(x)
