Question

Prove that :-

$\longrightarrow \sf \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta}$

Don't spam! ​

Answer 1

$\qquad \sf Prove \:that \:\::\: \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta}$

$\:\:\:\bigstar \underline {\bf According \:to \:the \:Question \::\:}$

$\dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta - \:( \:cosec^2 \theta- cot^2 \:\theta ) \:}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta - \: \bigg[ ( cosec \theta + cos \theta ) \:( \:cosec \theta- cot\:\theta ) \bigg] \:}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\\dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta \:\bigg[ 1 -\:( \:cosec \theta- cot\:\theta ) \bigg] \:}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta \:\bigg[ 1 \:- \:cosec \theta + cot\:\theta \bigg] \:}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cot\theta + cosec\theta \:\bigg[ \cancel { 1 \:- \:cosec \theta + cot\:\theta } \bigg] \:}{\cancel{ cot\theta - cosec\theta + 1} } = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: cot\theta + cosec\theta = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cos\theta }{sin \theta } + cosec\theta = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{cos\theta }{sin \theta } + \dfrac{1}{sin \theta} = \dfrac{1 + cos\theta}{sin\theta} \:\\\\ \dashrightarrow \sf \:\: \dfrac{1 + cos\theta }{sin \theta } = \dfrac{1 + cos\theta}{sin\theta} \:$

$\qquad \underline {\pmb{\bf Hence, \:Verified \:!\:}}$

Answer 2

To Prove :-

• $\sf \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1} = \dfrac{1 + cos\theta}{sin\theta}$

Proof:-

$\: \: \: \: \: \: \: \: \: \:\sf: \implies L.H.S$

$\: \: \: \: \: \: \: \pink{ \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - 1}{cot\theta - cosec\theta + 1}}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - \big(cosec^2\theta - cot^2\theta\big)}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta - \big(cosec\theta + cot\theta\big)\big(cosec\theta - cot\theta\big)}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta\Big(1 - \big(cosec\theta - cot\theta\big)\Big)}{cot\theta - cosec\theta + 1}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf \dfrac{cot\theta + cosec\theta\Big(1 - cosec\theta + cot\theta\Big)}{1 - cosec\theta + cot\theta}$

$\: \: \: \: \: \: \: \: \: \::\implies \sf cot\theta + cosec\theta$

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

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

$\: \: \: \: \: \: \: \: \: \:\sf: \implies R.H.S$

• Hence, Proved..!!