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Question

If $equals fraction numerator 2 pi over denominator 7 end fraction$ ,then tan $alpha tan invisible function application 2 alpha plus tan invisible function application 2 alpha tan invisible function application 4 alpha plus tan invisible function application 4 alpha tan invisible function application alpha equals$

$- 1$
$- 3$
$- 5$
$- 7$

hint Hint:

In this question, we have to find the value of tan $alpha tan invisible function application 2 alpha plus tan invisible function application 2 alpha tan invisible function application 4 alpha plus tan invisible function application 4 alpha tan invisible function application alpha$ , if $alpha$ $equals fraction numerator 2 pi over denominator 7 end fraction$ . For this we will solve the function using common trigonometric identity and simplify it and later substitute the value of $alpha$ in the equation.

The correct answer is: $negative 7$

$tan space alpha. tan space 2 alpha plus tan space 2 alpha. tan space 4 alpha plus tan space 4 alpha. tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction. fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction plus fraction numerator sin space 2 alpha over denominator cos space 2 alpha end fraction. fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction plus fraction numerator sin space 4 alpha over denominator cos space 4 alpha end fraction. fraction numerator sin space alpha over denominator cos space alpha end fraction equals fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus open parentheses 1 minus fraction numerator sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus open parentheses fraction numerator cos space alpha. cos space 2 alpha. cos space 4 alpha plus sin space alpha. sin space 2 alpha. cos space 4 alpha plus cos space alpha. sin space 2 alpha. sin space 4 alpha plus sin space alpha. cos space 2 alpha. sin space 4 alpha over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction close parentheses equals 1 minus fraction numerator cos space left parenthesis alpha plus 2 alpha plus 4 alpha right parenthesis over denominator cos space alpha. cos space 2 alpha. cos space 4 alpha end fraction equals 1 minus fraction numerator cos space 7 alpha over denominator begin display style fraction numerator sin space 8 alpha over denominator 8 space sin space alpha end fraction end style end fraction equals 1 minus fraction numerator 8. sin space alpha. cos space 7 alpha over denominator sin space 8 alpha end fraction S u b s t i t u t i n g space t h e space v a l u e space o f space alpha equals fraction numerator 2 pi over denominator 7 end fraction comma space equals 1 minus fraction numerator 8. sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction close parentheses. cos space 2 pi over denominator sin space open parentheses fraction numerator 2 pi over denominator 7 end fraction plus 2 pi close parentheses end fraction equals 1 minus 8 equals negative 7$

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