Prove that sin theta upon (1-cos theta )= cosec theta + cot theta

1.	Prove that sin theta upon (1-cos theta )= cosec theta + cot theta
Answer» $\\ \mathcal{\large{\underline {★ \: \: QUESTION\:\:}}} : - \begin{cases} \sf { \underbrace{ \overbrace{ \bf{Prove: - } \: \: \tt{ \dfrac{sin \: \theta}{1 - cos \: \theta} = cosec \: \theta \: + \: cot \: \theta}}}} \end{cases} \\\\\\$ ____________________________ $\\ \\\huge \underline{ \underline{ { \cal \: {★S}} \large \red{ \tt \: O} \green{ \tt \: L} \blue{ \tt \: <klux>U</klux>} \orange{ \tt \: T} \purple{ \tt \: I} \red{ \tt \: O} \blue{ \tt \: N}:-}} \\\\$ $\large {\underbrace{ \underline { \frak { \dag \:\:Finding \: \: \: \: \: L. \: H . \: S,}}}}\\\\$ $\\ \implies \sf \dfrac{sin \: \theta}{1 - \: cos \: \theta} \\\\$ $\large \underbrace{ \underline { \frak { \dag \: \: multiply \: numerator \: and \: dinominator \: by \: \red{\sf(1 + cos \: \theta).}}}}\\\\$ $: \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ (1 - cos \: \theta )(1 + cos \: \theta )} \\\\\ : \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ 1 - cos \: {}^{2} \: \theta } \\\\\ : \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ sin \: {}^{2} \: \theta } \\\\\ : \implies \tt \dfrac{1 + cos \: \theta}{ sin \: \: \theta } \\\\\ : \implies \tt \dfrac{1 }{ sin \: \: \theta } + \frac{cos \: \theta}{sin \: \theta} \\\\\ :\longmapsto \bf{cosec \: \theta \: + \: cot \: \theta \: = R.H.S}\\\\$ ☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯ $\\ \\\:\: \therefore \: \: \boxed{\large{\cal{LHS = RHS}}} \\ \\ \\ \large \tt\underline{ \underline{\overline{\overline{\mid{\mid{\red{ \lgroup Hence \: proved!! \rgroup}\mid}\mid}}}}}$ $\\\\$ ★ HOPE it HELPS u ★

Answer»

$\\ \mathcal{\large{\underline {★ \: \: QUESTION\:\:}}} : - \begin{cases} \sf { \underbrace{ \overbrace{ \bf{Prove: - } \: \: \tt{ \dfrac{sin \: \theta}{1 - cos \: \theta} = cosec \: \theta \: + \: cot \: \theta}}}} \end{cases} \\\\\\$

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$\\ \\\huge \underline{ \underline{ { \cal \: {★S}} \large \red{ \tt \: O} \green{ \tt \: L} \blue{ \tt \: <klux>U</klux>} \orange{ \tt \: T} \purple{ \tt \: I} \red{ \tt \: O} \blue{ \tt \: N}:-}} \\\\$

$\large {\underbrace{ \underline { \frak { \dag \:\:Finding \: \: \: \: \: L. \: H . \: S,}}}}\\\\$

$\\ \implies \sf \dfrac{sin \: \theta}{1 - \: cos \: \theta} \\\\$

$\large \underbrace{ \underline { \frak { \dag \: \: multiply \: numerator \: and \: dinominator \: by \: \red{\sf(1 + cos \: \theta).}}}}\\\\$

$: \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ (1 - cos \: \theta )(1 + cos \: \theta )} \\\\\ : \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ 1 - cos \: {}^{2} \: \theta } \\\\\ : \implies \tt \dfrac{sin \: \theta \: (1 + cos \: \theta)}{ sin \: {}^{2} \: \theta } \\\\\ : \implies \tt \dfrac{1 + cos \: \theta}{ sin \: \: \theta } \\\\\ : \implies \tt \dfrac{1 }{ sin \: \: \theta } + \frac{cos \: \theta}{sin \: \theta} \\\\\ :\longmapsto \bf{cosec \: \theta \: + \: cot \: \theta \: = R.H.S}\\\\$

☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯☯

$\\ \\\:\: \therefore \: \: \boxed{\large{\cal{LHS = RHS}}} \\ \\ \\ \large \tt\underline{ \underline{\overline{\overline{\mid{\mid{\red{ \lgroup Hence \: proved!! \rgroup}\mid}\mid}}}}}$

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