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

Question

3748 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-14T22:40:57

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

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) } #