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Question

The sum of the series, $sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times$ upto n terms, is

$\frac{1}{2} [\tan 3^{n} θ - \tan 3^{x - 1} θ]$
$[\tan 3^{n} θ - \tan θ]$
$\frac{1}{2} [\tan 3^{n} θ - \tan θ]$
$\frac{1}{2} (\tan 3^{n} θ - 1)$

hint Hint:

In the question we will simplify each term using trigonometric identities separately and then try to find any pattern in the series, then we will add the terms to get the sum of the series.

The correct answer is: $1 half open square brackets tan space 3 to the power of n theta minus tan space theta close square brackets$

Sn= $sin space theta sec space 3 theta plus sin space 3 theta sec space 3 squared theta plus sin space 3 squared theta sec space 3 cubed theta plus times times$ upto n terms.
$N o w comma space sin theta. s e c 3 theta equals fraction numerator sin theta over denominator cos 3 theta end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 sin theta. cos theta over denominator 2 cos 3 theta. cos theta end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half. fraction numerator sin 2 theta over denominator cos theta cos 3 theta end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin open parentheses 3 theta minus theta close parentheses over denominator cos theta. cos 3 theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin 3 theta cos theta minus cos 3 theta sin theta over denominator cos theta. cos 3 theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator sin 3 theta over denominator cos theta end fraction minus fraction numerator sin theta over denominator cos theta end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half open parentheses tan 3 theta minus tan theta close parentheses S i m i l a r l y comma space sin 3 theta. s e c 9 theta equals 1 half open parentheses tan 9 theta minus tan theta close parentheses comma space sin 9 theta. s e c 27 theta equals 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses comma space sin 3 to the power of open parentheses n minus 1 close parentheses end exponent theta. s e c 3 to the power of n theta equals 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses S o comma space S n equals 1 half open parentheses tan 3 theta minus tan theta close parentheses plus 1 half open parentheses tan 9 theta minus tan theta close parentheses plus 1 half open parentheses tan 27 theta minus tan 9 theta close parentheses plus..... plus 1 half open parentheses tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 theta minus tan theta plus tan 9 theta minus tan theta plus tan 27 theta minus tan 9 theta plus...... plus tan 3 to the power of n theta minus tan 3 to the power of open parentheses n minus 1 close parentheses end exponent theta close parentheses space space space space equals 1 half open parentheses tan 3 to the power of n theta minus tan theta close parentheses space space space space space space space space space space space space$

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