Question

trigno.... [jee advanced] ​

Answer 1

❏ Solution:-

$\sf\bf{\longrightarrow} \bf\sum\limits_{k=1}^{13}\frac{1}{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}] \sin[\frac{\pi}{4}+\frac{k\pi}{6}]}$

$\sf\longrightarrow \bf\sum\limits_{k=1}^{13}\frac{\frac{1}{ \frac{1}{2} } \times \frac{1}{2} }{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}] \sin[\frac{\pi}{4}+\frac{k\pi}{6}]}$

$\sf\longrightarrow \bf\frac{1}{\frac{1}{2}}\sum\limits_{k=1}^{13}\frac{\frac{1}{2}}{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}] \sin[\frac{\pi}{4}+\frac{k\pi}{6}]}$

$\sf\longrightarrow \bf2\sum\limits_{k=1}^{13}\frac{\sin\frac{\pi}{4}}{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}] \sin[\frac{\pi}{4}+\frac{k\pi}{6}]}$

$\sf\longrightarrow \bf 2\sum\limits_{k=1}^{13}\frac{\sin[\{\frac{\pi}{4}+\frac{k\pi}{6}\}-\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}]}{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}]\sin[\frac{\pi}{4}+\frac{k\pi}{6}]}$

$\small\sf\longrightarrow \bf 2\sum\limits_{k=1}^{13}\frac{\sin\{\frac{\pi}{4}+\frac{k\pi}{6}\}\cos\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}-\cos\{\frac{\pi}{4}+\frac{k\pi}{6}\}\sin\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}}{\sin[\frac{\pi}{4}+\frac{(k-1)\pi}{6}] \sin[\frac{\pi}{4}+\frac{k\pi }{6}]}$

$\sf\longrightarrow \bf 2\sum\limits_{k=1}^{13}[\frac{ \cancel{\sin\{\frac{\pi}{4}+\frac{k\pi}{6}\}}\cos\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}}{\sin\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\} \cancel{\sin\{\frac{\pi}{4}+\frac{k\pi}{6}\}}}-\frac{\cos\{\frac{\pi}{4}+\frac{k\pi}{6}\} \cancel{\sin\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}}}{ \cancel{\sin\{\frac{\pi}{4}+\frac{(k-1)\pi}{6}\}}\{\sin\frac{\pi}{4}+\frac{k\pi}{6}\}}]$

$\sf\longrightarrow \bf 2\sum\limits_{k=1}^{13}[\frac{ \cos \{ \frac{\pi}{4} + \frac{(k - 1)\pi}{6} \} }{ \sin \{ \frac{\pi}{4} + \frac{(k - 1)\pi}{6} \}} - \frac{\cos \{ \frac{\pi}{4} + \frac{k \pi}{6} \} }{\sin \{ \frac{\pi}{4} + \frac{k \pi}{6} \}} ]$

$\sf\longrightarrow \bf 2\sum\limits_{k=1}^{13}[ \cot\{ \frac{\pi}{4} + \frac{(k - 1)\pi}{6} \} - \cot \{ \frac{\pi}{4} + \frac{k\pi}{6} \} ]$

$\small\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4} + \frac{(1 - 1)\pi}{6} \} - \cot \{ \frac{\pi}{4} + \frac{\pi}{6} \} \\ + \cot \{ \frac{\pi}{4} + \frac{(2 - 1)\pi}{6} \} - \cot \{ \frac{\pi}{4} + \frac{2\pi}{6} \} \\ + \cot \{ \frac{\pi}{4} + \frac{(3- 1)\pi}{6} \} - \cot \{ \frac{\pi}{4} + \frac{3\pi}{6} \} \\ + ................................. \\ + \cot \{ \frac{\pi}{4} + \frac{(13- 1)\pi}{6} \} - \cot \{ \frac{\pi}{4} + \frac{13\pi}{6} \}]</p><p>$

$\small\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4}\} - \cancel{\cot \{ \frac{\pi}{4} + \frac{\pi}{6} \} } \\ + \cancel{\cot \{ \frac{\pi}{4} + \frac{\pi}{6} \}} - \cancel{\cot \{ \frac{\pi}{4} + \frac{2\pi}{6} \} }\\ + \cancel{\cot \{ \frac{\pi}{4} + \frac{2\pi}{6} \}} - \cancel{\cot \{ \frac{\pi}{4} + \frac{3\pi}{6} \}} \\ + ................................. \\ + \cancel{\cot \{ \frac{\pi}{4} + \frac{12\pi}{6} \} }- \cot \{ \frac{\pi}{4} + \frac{13\pi}{6} \}]$

$\sf \longrightarrow 2 \times [\cot \{ \frac{\pi}{4} \} - \cot \{ \frac{\pi}{4} + \frac{13\pi}{6} \}]$

$\sf \longrightarrow 2 \times [\cot \{ \frac{\pi}{4} \} - \cot \{ \frac{\pi}{4} + ( 2\pi + \frac{\pi}{6} )\}]$

$\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4} \}- \cot \{2\pi + ( \frac{\pi}{4} + \frac{\pi}{6} )\}]$

$\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4} \}- \cot \{ \frac{\pi}{4} + \frac{\pi}{6} \}]$

$[ \because \cot(2\pi + \theta) = \cot \theta \: \: ]$

$\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4} \}- \frac{1}{\tan\{ \frac{\pi}{4} + \frac{\pi}{6} \}}]$

$\sf\longrightarrow \bf 2 \times[ \cot \{ \frac{\pi}{4} \}- \frac{1 - \tan \frac{ \pi}{4} \tan \frac{\pi}{6} }{\tan\frac{\pi}{4} + \tan \frac{\pi}{6} }]$

$\sf\longrightarrow \bf 2 \times[1 - \frac{1 - \tan \frac{\pi}{6} }{1 + \tan \frac{\pi}{6} } ]$

$( \because \cot \frac{\pi}{4} = \tan \frac{\pi}{4} = 1)$

$\sf\longrightarrow \bf 2 \times[ \frac{ \cancel1 + \tan \frac{\pi}{6} - \cancel1 + \tan \frac{\pi}{6} }{1 + \tan \frac{\pi}{6} } ]$

$\sf\longrightarrow \bf 2 \times[ \frac{ 2\tan \frac{\pi}{6} }{1 + \tan \frac{\pi}{6} } ]$

$\sf\longrightarrow \bf [ \frac{ 4\tan \frac{\pi}{6} }{1 + \tan \frac{\pi}{6} } ]</p><p>$

$\sf\longrightarrow \bf [ \frac{ 4\tan \frac{\pi}{6} }{1 + \tan \frac{\pi}{6} } ] </p><p>$

$\sf\longrightarrow \bf [ \frac{ 4 \times \frac{1}{ \sqrt{3} } }{1 + \frac{1}{ \sqrt{3} } } ]$

$\sf\longrightarrow \bf [ \frac{ \frac{4}{ \cancel{\sqrt{3} } } }{ \frac{ \sqrt{3} + 1}{ \cancel{\sqrt{3} } }} ]$

$\sf\longrightarrow \bf [ \frac{ 4 }{ \sqrt{3} + 1} ]$

$\sf\longrightarrow \bf [ \frac{ 4 \times ( \sqrt{3} - 1) }{ ( \sqrt{3} + 1)( \sqrt{3} - 1)} ]$

$\sf\longrightarrow \bf [ \frac{ \cancel 4 \times ( \sqrt{3} - 1) }{ \cancel2 } ]$

$\sf\longrightarrow \bf \boxed{ \bf\large{\red{2( \sqrt{3} - 1) }}}$