tanα+2tan2α+4tan4α+8cot8α=
tanα
tan2α
cotα
cot2α
Explanation for the correct answer:
Simplifying the given expression:
tanα+2tan2α+4tan4α+8cot8α=tanα+2tan2α+4tan4α+81tan8α∵cotx=1tanx=tanα+2tan2α+4tan4α+81tan2(4α)=tanα+2tan2α+4tan4α+812tan(4α)1-tan2(4α)∵tan2A=2tanA(1-tan2A)=tanα+2tan2α+4tan4α+81-tan2(4α)2tan4α=tanα+2tan2α+4tan2(4α)+4-4tan2(4α)tan4α=tanα+2tan2α+41tan4α=tanα+2tan2α+41-tan2(2α2tan2α∵tan2A=2tanA(1-tan2A)=tanα+2tan2(2α)+2-2tan2(2α)tan2α=tanα+2tan2α=tanα+2(1-tan2α)2tanα∵tan2A=2tanA(1-tan2A)=tan2α+1-tan2αtanα=1tanα=cotα∵cotx=1tanx
Therefore, the correct answer is option (C).