How do you verify the identity #(cost+cos3t)/(sin3t-sint)=cott#?

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Answer 1 · 2017-03-08T13:42:29

See proof below

We use the sum to products formulae

#cosa+cosb=1/2cos((a+b)/2)cos((a-b)/2)#

#sina-sinb=1/2sin((a-b)/2)cos((a+b)/2)#

In our case

#a=3t#

and

#b=t#

Therefore,

#LHS=(cos3t+cost)/(sin3t-sint)#

#=(1/2cos((3t+t)/2)cos((3t-t)/2))/(1/2sin((3t-t)/2)cos((3t+t)/2))#

#=(cancel(cos2t)cost)/(sintcancel(cos2t))#

#=cost/sint#

#=cott#

#=RHS#

#QED#