Question

prove that : sin 0/cot 0 + cosec 0 =2+ sin 0/cot 0 - cosec 0​

Answer 1

$\boxed{ \underline{ \underline{\footnotesize\tt \purple{Used \: Formula \: to \: solve \: this \: question:-}}}} \: \footnotesize✺$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\purple\leadsto \footnotesize \tt{ {cosec}^{2} \: \theta \: - \: {cot }^{2} \: \theta \: = 1}$

$\purple\leadsto\footnotesize \tt{ cosec \: \theta = \frac{1}{sin \: \theta} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\boxed{ \underline{ \underline{\footnotesize\tt \purple{Solution:-}}}} \: \footnotesize✺$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\purple\leadsto \tt{ \frac{sin \: \theta}{cot \: \theta \: + \: cosec \: \theta} = \footnotesize2 \: + \: } \tt{ \frac{sin \: \theta}{cot \: \theta \: - \: cosec \: \theta} }$

$\purple\leadsto\tt{ \frac{sin \: \theta}{cot \: \theta \: + \: cosec \: \theta} - \frac{sin \: \theta}{cot \: \theta \: - \: cosec \: \theta} = \footnotesize2 }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\boxed{ \underline{ \underline{\footnotesize\tt \purple{Now \: solve \: LHS : - }}}}$

$\purple\leadsto\tt{ \frac{sin \: \theta}{cot \: \theta \: + \: cosec \: \theta} - \frac{sin \: \theta}{cot \: \theta \: - \: cosec \: \theta} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\boxed{ \underline{ \underline{\footnotesize\tt \purple{Same \: the \: denominator }}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\purple\leadsto\tt{ \frac{sin \: \theta \: (cot \: \theta \: - \: cosec \: \theta) \: - \: sin \: \theta \: (cot \: \theta \: + \: cosec \: \theta)}{(cot \: \theta \: + \: cosec \: \theta)(cot \: \theta \: - \: cosec \: \theta)}}$

$\purple\leadsto\tt{ \frac{sin \: \theta \: (cot \: \theta \: - \: cosec \: \theta \: - cot \: \theta \: - \: cosec \: \theta)}{{cot }^{2} \: \theta \: - \: {cosec}^{2} \: \theta} }$

$\purple\leadsto \tt{ \frac{sin \: \theta \: ( \cancel{cot \: \theta} \: - \: cosec \: \theta \: \cancel{ - cot \: \theta} \: - \: cosec \: \theta)}{{cot }^{2} \: \theta \: - \: {cosec}^{2} \: \theta} }$

$\purple\leadsto\tt{ \frac{sin \: \theta \: ( \: - \: cosec \: \theta \: \: - \: cosec \: \theta)}{{cot }^{2} \: \theta \: - \: {cosec}^{2} \: \theta} }$

$\purple\leadsto\tt{ \frac{sin \: \theta \: ( \: -2 \: cosec \: \theta)}{{cot }^{2} \: \theta \: - \: {cosec}^{2} \: \theta} }$

$\purple\leadsto\tt{ \frac{sin \: \theta \: ( \: -2 \: cosec \: \theta)}{ - \: ({cosec}^{2} \: \theta \: - \: {cot }^{2} \: \theta \: )} }$

$\purple\leadsto\tt{ \frac{sin \: \theta \: ( \: \cancel -2 \: cosec \: \theta)}{ \cancel - \: ({cosec}^{2} \: \theta \: - \: {cot }^{2} \: \theta \: )} }$

$\purple\leadsto \tt{ \frac{2 \: sin \: \theta \: ( \: cosec \: \theta)}{ \: ({cosec}^{2} \: \theta \: - \: {cot }^{2} \: \theta \: )} }$

$\purple\leadsto\tt{ \frac{2 \: sin \: \theta \: \times \frac{1}{sin \: \theta} }{ 1} }$

$\purple\leadsto\tt{ \frac{2 \cancel{\: sin \: \theta }\: \times \frac{1}{ \cancel{sin \: \theta}} }{ 1} }$

$\leadsto\boxed{ \underline{ \underline{\footnotesize\tt \purple{2 }}}} \: \footnotesize✺$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\footnotesize\tt{ L.H.S = R.H.S }$

$\footnotesize\tt{ Hence ,Proved }$

Answer 2

Answer is above

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