If cosec A=2–√, find the value of 2 sin2 A+3 cot2 A4(tan2 A−cos2 A)

Given: $\operatorname{cosec} A=\sqrt{2}$

We have to find the value of the expression $\frac{2 \sin ^{2} A+3 \cot ^{2} A}{4\left(\tan ^{2} A-\cos ^{2} A\right)}$

We know that,

$\operatorname{cosec} A=\sqrt{2}$

$\Rightarrow \sin A=\frac{1}{\operatorname{cosec} A}=\frac{1}{\sqrt{2}}$

$\cos A=\sqrt{1-\sin ^{2} A}=\sqrt{1-\left(\frac{1}{\sqrt{2}}\right)^{2}}=\frac{1}{\sqrt{2}}$

$\tan A=\frac{\sin A}{\cos A}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1$

$\cot A=\frac{1}{\tan A}=\frac{1}{1}=1$

Therefore,

$\frac{2 \sin ^{2} A+3 \cot ^{2} A}{4\left(\tan ^{2} A-\cos ^{2} A\right)}=\frac{2 \times\left(\frac{1}{\sqrt{2}}\right)^{2}+3 \times 1^{2}}{4\left(1^{2}-\left(\frac{1}{\sqrt{2}}\right)^{2}\right)}$

$=2$

Hence, the value of the given expression is 2.