{1/tan 3A+tanA }-{1/cot3A+cot A}
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Let us write t for tan A.
It may help to know that
tan 3A=(3t-t^3)/(1–3t^2)
tan 3A - tan A = (2t+2t^3)/(1–3t^2)
cot 3A = (1–3t^2)/(3t-t^3)
cot 3A-cot A=(1–3t^2)/(3t-t^3)-1/t
=(-2–2t^2)/(3t-t^3)
1/(tan3A-tan A)-1/(cot 3A - cot A)
=(1–3t^2)/(2t(1+t^2))
+(3t-t^3)/(2(1+t^2))
=(1-t^4)/(2t(1+t^2))
=(1-t^2)/(2t)
=cot 2A.
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