Question

If sin theta + cosec theta =2 then find sin^19theta +cosec^19theta =??
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Answer 1

Appropriate Question :

• If $\sin \theta + \cosec \theta = 2$ . Then , Find the value of $\sin^{19}\theta + \cosec ^{19} \theta$ .

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Given : $\sin \theta + \cosec \theta = 2$

Exigency to find : The Value of : $\sin^{19}\theta + \cosec ^{19} \theta$

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$\qquad \dag\:\:\bigg\lgroup \sf{ \sin \theta + \cosec \theta = 2 }\bigg\rgroup$

━━━━ Finding Value of : $\sin\theta \:\:\&\:\: \cosec \theta$

Now , By Solving the Given :

$\qquad \longmapsto \sf \sin \theta + \cosec \theta = 2$

$\dag\:\:\it{ As,\:We\:know\:that\::}$

• $\sf \cosec \theta = \dfrac{1}{\sin\theta }$

$\qquad \longmapsto \sf \sin \theta + \cosec \theta = 2$

$\qquad \longmapsto \sf \sin \theta + \dfrac{1}{\sin\theta } = 2$

$\qquad \longmapsto \sf \sin \theta + \dfrac{1}{\sin\theta } - 2 = 0$

$\qquad \longmapsto \sf \dfrac{\sin^2 \theta + 1 - 2\sin\theta}{\sin\theta } = 0$

$\qquad \longmapsto \sf \sin^2 \theta + 1 - 2\sin\theta = 0 \times \sin \theta$

$\qquad \longmapsto \sf \sin^2 \theta + 1 - 2\sin\theta = 0$

$\dag\:\:\it{ As,\:We\:know\:that\::}$

• $\sf (a - b)^2 = a^2 + b^2 - 2ab$

$\qquad \longmapsto \sf \sin^2 \theta + 1 - 2\sin\theta = 0$

$\qquad \longmapsto \sf (\sin \theta - 1 )^2 = 0$

$\qquad \longmapsto \sf \sin \theta - 1 = 0$

$\qquad \longmapsto \frak{\underline{\purple{\:\sin\theta = 1 }} }\bigstar$

$\dag\:\:\it{ As,\:We\:know\:that\::}$

• $\sf \cosec \theta = \dfrac{1}{\sin\theta }$

⠀⠀⠀⠀⠀Here, $\sin \theta = 1$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \longmapsto \sf \cosec \theta = \dfrac{1}{\sin\theta }$

$\qquad \longmapsto \sf \cosec \theta = \cancel{\dfrac{1}{1 }}$

$\qquad \longmapsto \frak{\underline{\purple{\:\cosec\theta = 1 }} }\bigstar$

━━━━ Finding Value of : $\sin^{19}\theta + \cosec ^{19} \theta$

⠀⠀⠀⠀⠀Here $\sin \theta = 1 \:\:\:\& \:\:\:\: \cosec \theta = 1$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \longmapsto \sf \sin^{19}\theta + \cosec ^{19} \theta$

$\qquad \longmapsto \sf 1^{19} + 1 ^{19}$

$\qquad \longmapsto \sf 1 + 1$

$\qquad \longmapsto \bf \bigg( \:\:2 \:\:\bigg)\qquad \longrightarrow \:\: Required \:AnswEr \:$

Therefore,

⠀⠀⠀⠀⠀$\therefore {\underline{ \mathrm {\:The\:value \:of\: \sin^{19}\theta + \cosec ^{19} \theta \:is\:\bf{2}}}}$

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