Question

See the image and plss answer the question ​

Answer 1

$\sf{ \bold{ To \: Prove \: - }} \\ \\ \sf{ \dfrac{ \sin { \theta } }{ \cot { \theta } + \csc{ \theta } } - \dfrac{ \sin { \theta } }{ \cot { \theta } - \csc{ \theta } } = 2 } \\ \\ \sf{ \implies { \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } + \frac{1}{ \sin { \theta }} } - \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } - \frac{1}{ \sin { \theta }} } = 2 }} \\ \\ \sf{ \bold { Taking \: LCM \: - }} \\ \\ \sf{ \implies { \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } + 1 }{ \sin { \theta } } } - \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } - 1 }{ \sin { \theta } } } = 2 }} \\ \\ \sf{ \implies { \dfrac{ sin^2 { \theta } }{ 1 + \cos { \theta } } - \dfrac{ sin^2 { \theta } }{ 1 - \cos { \theta } } = 2 }} \\ \\ \sf{ \bold { sin^2 { \theta } = 1 - cos^2 { \theta } }} \\ \\ \sf{ \bold{ Substituting \: this \: value \: - }} \\ \\ \sf{ \implies { 1 - \cos { \theta } + 1 + \cos{ \theta } }} \\ \\ \sf{ \implies { 2 }} \\ \\ \sf{ \bold{ Hence \: Proved } }$

$\sf{ \bold{ To \: Prove \: - }} \\ \\ \sf{ \dfrac{ \sin { \theta } }{ \cot { \theta } + \csc{ \theta } } - \dfrac{ \sin { \theta } }{ \cot { \theta } - \csc{ \theta } } = 2 } \\ \\ \sf{ \implies { \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } + \frac{1}{ \sin { \theta }} } - \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } - \frac{1}{ \sin { \theta }} } = 2 }} \\ \\ \sf{ \bold { Taking \: LCM \: - }} \\ \\ \sf{ \implies { \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } + 1 }{ \sin { \theta } } } - \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } - 1 }{ \sin { \theta } } } = 2 }} \\ \\ \sf{ \implies { \dfrac{ sin^2 { \theta } }{ 1 + \cos { \theta } } - \dfrac{ sin^2 { \theta } }{ 1 - \cos { \theta } } = 2 }} \\ \\ \sf{ \bold { sin^2 { \theta } = 1 - cos^2 { \theta } }} \\ \\ \sf{ \bold{ Substituting \: this \: value \: - }} \\ \\ \sf{ \implies { 1 - \cos { \theta } + 1 + \cos{ \theta } }} \\ \\ \sf{ \implies { 2 }} \\ \\ \sf{ \bold{ Hence \: Proved } }$

$\sf{ \bold{ To \: Prove \: - }} \\ \\ \sf{ \dfrac{ \sin { \theta } }{ \cot { \theta } + \csc{ \theta } } - \dfrac{ \sin { \theta } }{ \cot { \theta } - \csc{ \theta } } = 2 } \\ \\ \sf{ \implies { \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } + \frac{1}{ \sin { \theta }} } - \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } - \frac{1}{ \sin { \theta }} } = 2 }} \\ \\ \sf{ \bold { Taking \: LCM \: - }} \\ \\ \sf{ \implies { \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } + 1 }{ \sin { \theta } } } - \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } - 1 }{ \sin { \theta } } } = 2 }} \\ \\ \sf{ \implies { \dfrac{ sin^2 { \theta } }{ 1 + \cos { \theta } } - \dfrac{ sin^2 { \theta } }{ 1 - \cos { \theta } } = 2 }} \\ \\ \sf{ \bold { sin^2 { \theta } = 1 - cos^2 { \theta } }} \\ \\ \sf{ \bold{ Substituting \: this \: value \: - }} \\ \\ \sf{ \implies { 1 - \cos { \theta } + 1 + \cos{ \theta } }} \\ \\ \sf{ \implies { 2 }} \\ \\ \sf{ \bold{ Hence \: Proved } }$

$\sf{ \bold{ To \: Prove \: - }} \\ \\ \sf{ \dfrac{ \sin { \theta } }{ \cot { \theta } + \csc{ \theta } } - \dfrac{ \sin { \theta } }{ \cot { \theta } - \csc{ \theta } } = 2 } \\ \\ \sf{ \implies { \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } + \frac{1}{ \sin { \theta }} } - \dfrac{ \sin{ \theta } }{ \frac{ \cos { \theta } }{ \sin { \theta } } - \frac{1}{ \sin { \theta }} } = 2 }} \\ \\ \sf{ \bold { Taking \: LCM \: - }} \\ \\ \sf{ \implies { \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } + 1 }{ \sin { \theta } } } - \dfrac{ \sin { \theta } }{ \frac{ \cos { \theta } - 1 }{ \sin { \theta } } } = 2 }} \\ \\ \sf{ \implies { \dfrac{ sin^2 { \theta } }{ 1 + \cos { \theta } } - \dfrac{ sin^2 { \theta } }{ 1 - \cos { \theta } } = 2 }} \\ \\ \sf{ \bold { sin^2 { \theta } = 1 - cos^2 { \theta } }} \\ \\ \sf{ \bold{ Substituting \: this \: value \: - }} \\ \\ \sf{ \implies { 1 - \cos { \theta } + 1 + \cos{ \theta } }} \\ \\ \sf{ \implies { 2 }} \\ \\ \sf{ \bold{ Hence \: Proved } }$

Answer 2

$iam \: representing \: theta \: with \: \alpha \\ \\ = > \frac{ \sin \alpha }{( \cot\alpha + \csc \alpha ) } + \frac{ \sin \alpha }{ (\cot \alpha - \csc\alpha) \ \ } \\ \\ = \frac{ \sin\alpha( \cot\alpha - \ \csc \alpha) - \sin \alpha( \cot \alpha + \csc \alpha) }{( \cot\alpha + \ \csc \alpha)( \cot\alpha - \ \csc \alpha) } \\ \\ = \frac{ \cos\alpha - \sin\alpha \ \csc \alpha - \cos\alpha - \sin\alpha \ \csc \alpha }{ \cot {}^{2} { \alpha } - \tan {}^{2} \alpha } \\ \\ = \frac{ - 2 }{ - 1} = 2 = rhs \\ \\ the \: formulas \: which \: i \: have \: used \: in \: this \: \\ \\ problem \: solvation \: are \: in \: the \: above \: pic$

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