If cosec A = $$\frac 54$$​, then the value of sin2A + 2cos2A is:

Correct option is B

Given
cosec A = $$\frac{5}{4}$$​​

Formula Used
​$$sin A = \frac{1}{cosec A} \\\ \\sin 2A = 2sin A cos A\\cos 2A = cos^2 A - sin^2 A\\$$​​
Solution
Since cosec A = \frac{5}{4}, we have sin A = \frac{4}{5}.
Using Pythagoras theorem, cos A = \frac{3}{5}.
Calculate sin 2A = $$2 × \frac{4}{5} × \frac{3}{5} = \frac{24}{25}$$​​
Calculate cos 2A = $$(\frac{3}{5})^2 - (\frac{4}{5})^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25}$$​​
Now, find sin 2A + 2cos 2A:
$$= \frac{24}{25} + 2(-\frac{7}{25}) \\\ \\= \frac{24}{25} - \frac{14}{25}\\\ \\= \frac{10}{25} = \frac{2}{5}\\\ \\$$

$$= \frac{24}{25} + 2(-\frac{7}{25}) \\\ \\= \frac{24}{25} - \frac{14}{25}\\\ \\= \frac{10}{25} = \frac{2}{5}$$

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Final Answer
So the correct answer is (b)