Question

If cosec A = √2, then find the value of :


$\boxed{ \sf \: \frac{2 {sin}^{2} A + 3 {cot}^{2}A }{4 {tan}^{2} A - {cos}^{2}A } }$
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Answer 1

Aиѕωєr —

$\qquad{\pmb{\bf{\leadsto \dfrac{8}{7}}}}$

Giνєи —

• Cosec A = √2,

Tσ Fiиd —

• ${ \sf \: \dfrac{2 {sin}^{2} A + 3 {cot}^{2}A }{4 {tan}^{2} A - {cos}^{2}A } }$

Sσℓυтiσи –

• We have , cosec A =√2

Tнєrєfσrє —

✪ sinA :

$\quad\rm{\implies sinA =\dfrac{1}{cosecA}}$

$\quad{\pmb{\rm{\implies sinA =\dfrac{1}{\sqrt{2}}}}}$

✪ cosA :

$\quad\rm{\implies cosA =\sqrt{1-sin^2 A}}$

$\quad\rm{\implies cosA = \sqrt{1-\left(\dfrac{1}{\sqrt{2}}\right)^2} }$

$\quad\rm{\implies cosA =\sqrt{1-\dfrac{1}{2}}}$

$\quad\rm{\implies cosA =\sqrt{\dfrac{1}{2}} }$

$\quad{\pmb{\rm{\implies cosA =\dfrac{1}{\sqrt{2}}}}}$

✪ tanA :

$\quad\rm{\implies tanA =\dfrac{sinA}{cosA}}$

$\quad\rm{\implies tanA =\dfrac{\cancel{\dfrac{1}{\sqrt{2}}}}{\cancel{\dfrac{1}{\sqrt{2}}}}}$

$\quad{\pmb{\rm{\implies tanA = 1}} }$

✪ cotA :

$\quad\rm{\implies cotA =\dfrac{1}{tanA}}$

$\quad\rm{\implies cotA = \dfrac{1}{1}}$

$\quad{\pmb{\rm{\implies cotA =1}}}$

Hєиcє —

$\quad\rm{\longrightarrow \dfrac{2sin^2 A + 3 cot^2 A }{4tan^2 A - cos ^2 A} }$

$\quad\rm{\longrightarrow \dfrac{2\times \left(\dfrac{1}{\sqrt{2}}\right)^2+3(1)^2}{4(1)^2 -\left(\dfrac{1}{\sqrt{2}}\right)^2}}$

$\quad\rm{\longrightarrow \dfrac{2\times \dfrac{1}{2} +3}{4-\dfrac{1}{2}}}$

$\quad\rm{\longrightarrow \dfrac{1+3}{\dfrac{7}{2}} }$

$\quad\rm{\longrightarrow \dfrac{4\times 2}{7} }$

$\quad{\pmb{\rm{\longrightarrow \dfrac{8}{7} }}}$