Question

cos theta. cosec theta - sin theta. sec theta / cosec theta + sin theta = codec theta. sec theta ​

Answer 1

$\underline\mathfrak\pink{To \: Proved:-}$

• $\: \: \: \: \: \leadsto \: \: \frac{Cos \theta \: Cosec \theta \: - \: Sin \theta \: Sec \theta}{Cos \theta \: + \: Sin \theta} \: \: = \: \: Cosec \theta \: - \: Sec \theta$

$\large\underline\mathfrak\pink{Solutions:-}$

LHS:-

$\: \: \: \: \: \leadsto \: \: \frac{Cos \theta \: Cosec \theta \: - \: Sin \theta \: Sec \theta}{Cos \theta \: + \: Sin \theta}$

$\: \: \: \: \: \leadsto \: \: \frac{Cos \theta \: {(\frac{1}{Sin \theta})} \: - \: Sin \theta \: {(\frac{1}{Cos \theta})}}{Cos \theta \: + \: Sin \theta} \: \: \: \: \: \: {[Sec \theta \: \: = \: \: \frac{1}{Cos \theta} \: \: and \: \: Cosec \: \: = \: \: \frac{1}{Sin \theta}]}$

$\: \: \: \: \: \leadsto \: \: \frac{{(\frac{Cos \theta}{Sin \theta})} \: - \: {(\frac{Sin \: theta}{Cos \theta})}}{Cos \theta \: + \: Sin \theta}$

$\: \: \: \: \: \leadsto \: \: \frac{\frac{{Cos}^{2} \theta \: - \: {Sin}^{2} \theta}{Sin \theta \: Cos \theta}}{Cos \theta \: + \: Sin \theta}$

$\: \: \: \: \: \leadsto \: \: \frac{\frac{{({Cos} \theta \: - \: {Sin} \theta)} \: {({Cos} \theta \: + \: {Sin} \theta)}}{Sin \theta \: Cos \theta}}{Cos \theta \: + \: Sin \theta} \: \: \: \: \: {[{a}^{2} \: - \: {b}^{2} \: \: = \: \: {(a \: + \: b)} \: \: {(a \: - \: b)}]}$

$\: \: \: \: \: \leadsto \: \: \frac{Cos \theta \: - \: Sin \theta}{Sin \theta \: Cos \theta}$

$\: \: \: \: \: \leadsto \: \: \frac{Cos \theta}{Sin \theta \: Cos \theta} \: - \: \frac{Sin \theta}{Sin \theta \: Cos \theta}$

$\: \: \: \: \: \leadsto \: \: \frac{1}{Sin \theta} \: - \: \frac{1}{Cos \theta}$

$\: \: \: \: \: \leadsto \: \: Cosec \theta \: - \: Sec \theta$

$\: \: \: \: \: \: \: RHS \: \: \leadsto \: \: Cosec \theta \: - \: Sec \theta$

• Proved

Answer 2

Answer:

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