Question

1+ cos 105° + cos 165°/sin 105° + sin 375°
= 0​

Answer 1

$\sf \bf \huge {\boxed {\mathbb {QUESTION}}}$

$\sf \bf 1+\dfrac{cos\:{105}^{\circ} +cos\:{165}^{\circ}}{sin\:{105}^{\circ} +sin\:{375}^{\circ}} =0$

$\sf \bf \huge {\boxed {\mathbb {ANSWER}}}$

$\sf \bf\implies 1+\dfrac{cos\:{105}^{\circ} +cos\:{165}^{\circ}}{sin\:{105}^{\circ} +sin\:{375}^{\circ}} =0$

$\sf \bf RHS$

$\sf \bf\implies 1+\dfrac{cos\:{105}^{\circ} +cos\:{165}^{\circ}}{sin\:{105}^{\circ} +sin\:{375}^{\circ}}$

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$\sf \bf cos\:{105}^{\circ} \:can\:be\:written \:as\::-$

$\sf \bf \implies cos\:\big({90}^{\circ} +{15}^{\circ} \big)$

$\implies {\boxed {\color {green} {\sf \bf - sin\:{15}^{\circ}}}}$

$\sf \bf cos\:{165}^{\circ} \:can\:be\:written \:as\::-$

$\sf \bf \implies cos\:\big({180}^{\circ} +{15}^{\circ} \big)$

$\implies {\boxed {\color {green} {\sf \bf - cos\:{15}^{\circ}}}}$

$\sf \bf sin\:{105}^{\circ} \:can\:be\:written \:as\::-$

$\sf \bf \implies sin\:\big({90}^{\circ} +{15}^{\circ} \big)$

$\implies {\boxed {\color {green} {\sf \bf cos\:{15}^{\circ}}}}$

$\sf \bf sin\:{375}^{\circ} \:can\:be\:written \:as\::-$

$\sf \bf \implies sin\:\big({360}^{\circ} +{15}^{\circ} \big)$

$\implies {\boxed {\color {green} {\sf \bf sin\:{15}^{\circ}}}}$

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${\underbrace {\overbrace {\color {orange} {\sf \bf Substitute \:the\:values}}}}$

$\sf \bf\implies 1+\dfrac{-sin\:{15}^{\circ} - cos\:{15}^{\circ}}{cos\:{15}^{\circ} +sin\:{15}^{\circ}}$

$\sf \bf\implies 1+\dfrac{-\big[sin\:{15}^{\circ} + cos\:{15}^{\circ}\big]}{cos\:{15}^{\circ} +sin\:{15}^{\circ}}$

$\sf \bf\implies 1+\dfrac{-\cancel {\big[sin\:{15}^{\circ} + cos\:{15}^{\circ}\big]}}{\cancel {cos\:{15}^{\circ} +sin\:{15}^{\circ}}}$

$\sf \bf\implies 1+\dfrac{-1}{1}$

$\sf \bf\implies 1+\big(-1\big)$

$\sf \bf\implies 1-1$

$\implies{\boxed {\boxed {\color {blue} {\sf \bf 0}}}}$

$\sf \bf LHS$

${\color {purple} \underline {\tt \therefore LHS=RHS}}$

$\sf \bf \huge {\boxed {\mathbb {HOPE \:IT \:HELPS \:YOU}}}$

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$\sf \bf \huge {\boxed {\mathbb {EXTRA\:INFORMATION}}}$

${\color {red} {\tt Identities}}$

$\sf \bf {sin}^{2}\:\theta+{cos}^{2}\:\theta =1$

$\sf \bf {tan}^{2}\:\theta+1={sec}^{2}\:\theta$

$\sf \bf {cot}^{2}\:\theta+1={cosec}^{2}\:\theta$

${\mathbb{\colorbox {orange} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {lime} {\boxed{\boxed{\boxed{\boxed{\boxed{\colorbox {aqua} {@suraj5070}}}}}}}}}}}}}}}$