Q.prove that
   tan x + 2 tan 2x+ 4 tan 4x + 8 cot 8x =cot x
 
 

Asked by imabhi264 | 24th May, 2017, 12:03: PM

Expert Answer:

begin mathsize 18px style Now comma space tany minus space cot space straight y
equals space tany minus fraction numerator 1 over denominator tan space straight y end fraction
space equals fraction numerator tan squared space straight y minus 1 over denominator tan space straight y end fraction
equals 2 open parentheses fraction numerator tan squared space straight y minus 1 over denominator 2 tan space straight y end fraction close parentheses
equals negative 2 open parentheses fraction numerator 1 minus tan squared space straight y over denominator 2 tan space straight y end fraction close parentheses
equals negative 2 open parentheses fraction numerator fraction numerator 1 over denominator 2 tan space straight y end fraction over denominator 1 minus tan squared space straight y end fraction close parentheses
equals negative 2 cross times fraction numerator 1 over denominator 2 tan squared space straight y end fraction
equals negative 2 cot space 2 straight y space space space space..... space left parenthesis straight i right parenthesis
LHS
equals tan space straight x space plus space 2 space tan space 2 straight x plus space 4 space tan space 4 straight x space plus space 8 space cot space 8 straight x
equals tan space straight x space minus cot space straight x plus space 2 space tan space 2 straight x plus space 4 space tan space 4 straight x space plus space 8 space cot space 8 straight x plus cot space straight x
equals negative 2 cot space 2 straight x plus space 2 space tan space 2 straight x plus space 4 space tan space 4 straight x space plus space 8 space cot space 8 straight x plus cot space straight x space space space space space space space..... left parenthesis from space left parenthesis straight i right parenthesis right parenthesis
equals 2 left parenthesis tan 2 straight x minus cot 2 straight x right parenthesis plus space 4 space tan space 4 straight x space plus space 8 space cot space 8 straight x plus cot space straight x
equals 2 left parenthesis negative 2 cot space 4 straight x right parenthesis plus space 4 space tan space 4 straight x plus 8 cot space 8 straight x plus cot space straight x
equals 4 left parenthesis tan 4 straight x minus cot space 4 straight x right parenthesis plus space 8 space cot space 8 straight x plus cot space straight x
equals 4 left parenthesis negative 2 cot space 8 straight x right parenthesis plus 8 cot space 8 straight x plus cot space straight x
equals negative 8 cot space 8 straight x plus 8 cot space 8 straight x plus cot space straight x
equals cot space straight x
equals RHS end style

Answered by Rebecca Fernandes | 24th Nov, 2017, 12:27: PM

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