Prove? cotx = sin5x+sin7x / cos5x-cos7x

2 Answers
Mar 21, 2018

See the proof below

Explanation:

Reminder

#sina+sinb=2sin((a+b)/2)cos((a-b)/2)#

#cosa-cosb=-2sin((a+b)/2)sin((a-b)/2)#

Therefore,

#RHS=(sin5x+sin7x)/(cos5x-cos7x)#

#=(2sin((5x+7x)/2)cos((5x-7x)/2))/(-2sin((5x+7x)/2)sin((5x-7x)/2))#

#=(cos((5x-7x)/2))/(-sin((5x-7x)/2))#

#=cosx/(-(-sinx))#

#=cotx#

#=LHS#

#QED#

Mar 21, 2018

Please see below.

Explanation:

WE have,
#color(red)((1)sinC+sinD=2sin((C+D)/2)cos((C-D)/2)#

#color(red)((2)cosC-cosD=-2sin((C+D)/2)sin((C-D)/2)#
Here,
#cotx=(sin5x+sin7x)/(cos5x-cos7x)#

#RHS=(sin5x+sin7x)/(cos5x-cos7x)#

Using (1) and (2),

#=(cancel(2)cancelsin((5x+7x)/2)cos((5x-7x)/2))/(-cancel2cancelsin((5x+7x)/2)sin((5x-7x)/2))#

#=-cos((5x-7x)/2)/sin((5x-7x)/2)#

#=-cos((-2x)/2)/sin((-2x)/2)#

#=-cos(-x)/sin(-x)#

#=-cosx/(-sinx#

#=cotx#