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,

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

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

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

$=cost/sint$

$=cott$

$=RHS$

$QED$

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