If $$(2 \sin A + \cosec A) = 2 \sqrt{2}$$, $$0^\circ < A < 90^\circ$$ then the value of $$2(\sin^{4}A + \cos^{4}A)$$ is:

$$(2 \sin A + \cosec A) = 2 \sqrt{2}$$

To find the value A, we satisfy the above equation so put the value of A = 45$$\degree$$

$$(2 \sin 45 \degree + \cosec 45 \degree) = 2 \sqrt{2}$$

$$(2 \times \frac{1}{\sqrt{2}} + \sqrt{2}) = 2 \sqrt{2}$$

$$ 2\sqrt{2} = 2\sqrt{2}$$

$$2(\sin^{4}A + \cos^{4}A)$$

= $$2(\sin^{4}45 \degree + \cos^{4}45\degree)$$

= $$2((\frac{1}{\sqrt{2}})^{4} + (\frac{1}{\sqrt{2}})^{4})$$

= $$2(\frac{1}{4} + \frac{1}{4}) = 2(\frac{1}{2}) = 1$$