If cot3⁡α+cot2⁡α+cot⁡α=1, then

cot3⁡α+cot2⁡α+cot⁡α=1⇒tan3⁡α−tan2⁡α−tan⁡α−1=0⇒tan⁡αtan2⁡α−1=1+tan2⁡α⇒tan⁡α=−1+tan2⁡α1−tan2⁡α⇒cos⁡2α⋅tan⁡α=−1Now cos⁡2α−tan⁡2α=−1tan⁡α−2tan⁡α1−tan2⁡α=−1−tan2⁡α+2tan2⁡αtan⁡α1−tan2⁡α=−1+tan2⁡αtan⁡α1−tan2⁡α=−1tan⁡αcos⁡2α=1