Question

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Answer 1

Step-by-step explanation:

$\rm \: cosec \theta = \sqrt{10}$

$\rm \because \: sin \theta = \frac{1}{ cosec \theta}$

$\rm \therefore \: sin \theta = \frac{1}{ \sqrt{10} }$

$\rm \because \: {cos}^{2} \theta = 1 - {sin}^{2} \theta$

$\rm \therefore \: {cos}^{2} \theta = 1 - {( \frac{1}{ \sqrt{10} } )}^{2}$

$\rm {cos}^{2} \theta = 1 - \frac{1}{10}$

$\rm {cos}^{2} \theta = \frac{10 - 1}{10} \: \:$

$\rm {cos}^{2} \theta = \frac{9}{10}$

$\rm cos \theta = \sqrt{ \frac{9}{10} }$

$\rm cos \theta = \frac{3}{ \sqrt{10} }$

$\rm \because \: \: sec \theta = \frac{1}{cos \theta}$

$\rm \therefore \: sec \theta = \frac{1}{ \frac{3}{\sqrt{10}} }$

$\rm \: sec \theta = \frac{ \sqrt{10} }{3}$

$\rm \because \: \: tan \theta = \frac{sin \theta}{cos \theta}$

$\rm \therefore \: \: tan \theta = \frac{ \frac{1}{ \sqrt{10} } }{ \frac{3}{ \sqrt{10} } }$

$\rm tan \theta = \frac{1}{ \cancel{ \sqrt{10} }} \times \frac{\cancel{ \sqrt{10} } }{3}$

$\rm tan \theta = \frac{1}{3}$

$\rm\because \: \: cot \theta = \frac{1}{ tan \theta}$

$\rm\therefore \: \: cot \theta = \frac{1}{ \frac{1}{3} }$

$\rm cot \theta = 3$

Answer 2

Step-by-step explanation:

$\rm \: cosec \theta = \sqrt{10}$

$\rm \because \: sin \theta = \frac{1}{ cosec \theta}$

$\rm \therefore \: sin \theta = \frac{1}{ \sqrt{10} }$

$\rm \because \: {cos}^{2} \theta = 1 - {sin}^{2} \theta$

$\rm \therefore \: {cos}^{2} \theta = 1 - {( \frac{1}{ \sqrt{10} } )}^{2}$

$\rm {cos}^{2} \theta = 1 - \frac{1}{10}$

$\rm {cos}^{2} \theta = \frac{10 - 1}{10} \: \:$

$\rm {cos}^{2} \theta = \frac{9}{10}$

$\rm cos \theta = \sqrt{ \frac{9}{10} }$

$\rm cos \theta = \frac{3}{ \sqrt{10} }$

$\rm \because \: \: sec \theta = \frac{1}{cos \theta}$

$\rm \therefore \: sec \theta = \frac{1}{ \frac{3}{\sqrt{10}} }$

$\rm \: sec \theta = \frac{ \sqrt{10} }{3}$

$\rm \because \: \: tan \theta = \frac{sin \theta}{cos \theta}$

$\rm \therefore \: \: tan \theta = \frac{ \frac{1}{ \sqrt{10} } }{ \frac{3}{ \sqrt{10} } }$

$\rm tan \theta = \frac{1}{ \cancel{ \sqrt{10} }} \times \frac{\cancel{ \sqrt{10} } }{3}$

$\rm tan \theta = \frac{1}{3}$

$\rm\because \: \: cot \theta = \frac{1}{ tan \theta}$

$\rm\therefore \: \: cot \theta = \frac{1}{ \frac{1}{3} }$

$\rm cot \theta = 3$

Answer 3

Step-by-step explanation:

$\rm \: cosec \theta = \sqrt{10}$

$\rm \because \: sin \theta = \frac{1}{ cosec \theta}$

$\rm \therefore \: sin \theta = \frac{1}{ \sqrt{10} }$

$\rm \because \: {cos}^{2} \theta = 1 - {sin}^{2} \theta$

$\rm \therefore \: {cos}^{2} \theta = 1 - {( \frac{1}{ \sqrt{10} } )}^{2}$

$\rm {cos}^{2} \theta = 1 - \frac{1}{10}$

$\rm {cos}^{2} \theta = \frac{10 - 1}{10} \: \:$

$\rm {cos}^{2} \theta = \frac{9}{10}$

$\rm cos \theta = \sqrt{ \frac{9}{10} }$

$\rm cos \theta = \frac{3}{ \sqrt{10} }$

$\rm \because \: \: sec \theta = \frac{1}{cos \theta}$

$\rm \therefore \: sec \theta = \frac{1}{ \frac{3}{\sqrt{10}} }$

$\rm \: sec \theta = \frac{ \sqrt{10} }{3}$

$\rm \because \: \: tan \theta = \frac{sin \theta}{cos \theta}$

$\rm \therefore \: \: tan \theta = \frac{ \frac{1}{ \sqrt{10} } }{ \frac{3}{ \sqrt{10} } }$

$\rm tan \theta = \frac{1}{ \cancel{ \sqrt{10} }} \times \frac{\cancel{ \sqrt{10} } }{3}$

$\rm tan \theta = \frac{1}{3}$

$\rm\because \: \: cot \theta = \frac{1}{ tan \theta}$

$\rm\therefore \: \: cot \theta = \frac{1}{ \frac{1}{3} }$

$\rm cot \theta = 3$

Answer 4

Step-by-step explanation:

$\rm \: cosec \theta = \sqrt{10}$

$\rm \because \: sin \theta = \frac{1}{ cosec \theta}$

$\rm \therefore \: sin \theta = \frac{1}{ \sqrt{10} }$

$\rm \because \: {cos}^{2} \theta = 1 - {sin}^{2} \theta$

$\rm \therefore \: {cos}^{2} \theta = 1 - {( \frac{1}{ \sqrt{10} } )}^{2}$

$\rm {cos}^{2} \theta = 1 - \frac{1}{10}$

$\rm {cos}^{2} \theta = \frac{10 - 1}{10} \: \:$

$\rm {cos}^{2} \theta = \frac{9}{10}$

$\rm cos \theta = \sqrt{ \frac{9}{10} }$

$\rm cos \theta = \frac{3}{ \sqrt{10} }$

$\rm \because \: \: sec \theta = \frac{1}{cos \theta}$

$\rm \therefore \: sec \theta = \frac{1}{ \frac{3}{\sqrt{10}} }$

$\rm \: sec \theta = \frac{ \sqrt{10} }{3}$

$\rm \because \: \: tan \theta = \frac{sin \theta}{cos \theta}$

$\rm \therefore \: \: tan \theta = \frac{ \frac{1}{ \sqrt{10} } }{ \frac{3}{ \sqrt{10} } }$

$\rm tan \theta = \frac{1}{ \cancel{ \sqrt{10} }} \times \frac{\cancel{ \sqrt{10} } }{3}$

$\rm tan \theta = \frac{1}{3}$

$\rm\because \: \: cot \theta = \frac{1}{ tan \theta}$

$\rm\therefore \: \: cot \theta = \frac{1}{ \frac{1}{3} }$

$\rm cot \theta = 3$

Answer 5

Step-by-step explanation:

$\rm \: cosec \theta = \sqrt{10}$

$\rm \because \: sin \theta = \frac{1}{ cosec \theta}$

$\rm \therefore \: sin \theta = \frac{1}{ \sqrt{10} }$

$\rm \because \: {cos}^{2} \theta = 1 - {sin}^{2} \theta$

$\rm \therefore \: {cos}^{2} \theta = 1 - {( \frac{1}{ \sqrt{10} } )}^{2}$

$\rm {cos}^{2} \theta = 1 - \frac{1}{10}$

$\rm {cos}^{2} \theta = \frac{10 - 1}{10} \: \:$

$\rm {cos}^{2} \theta = \frac{9}{10}$

$\rm cos \theta = \sqrt{ \frac{9}{10} }$

$\rm cos \theta = \frac{3}{ \sqrt{10} }$

$\rm \because \: \: sec \theta = \frac{1}{cos \theta}$

$\rm \therefore \: sec \theta = \frac{1}{ \frac{3}{\sqrt{10}} }$

$\rm \: sec \theta = \frac{ \sqrt{10} }{3}$

$\rm \because \: \: tan \theta = \frac{sin \theta}{cos \theta}$

$\rm \therefore \: \: tan \theta = \frac{ \frac{1}{ \sqrt{10} } }{ \frac{3}{ \sqrt{10} } }$

$\rm tan \theta = \frac{1}{ \cancel{ \sqrt{10} }} \times \frac{\cancel{ \sqrt{10} } }{3}$

$\rm tan \theta = \frac{1}{3}$

$\rm\because \: \: cot \theta = \frac{1}{ tan \theta}$

$\rm\therefore \: \: cot \theta = \frac{1}{ \frac{1}{3} }$

$\rm cot \theta = 3$