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If $\frac{3 π}{2} < A < 2 π$ then $\sin A = - 7 / 25$ then,

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\sin 2 A = - \frac{336}{625}

\cos \frac{A}{2} = \frac{\sqrt{2}}{5}

\tan \frac{A}{2} = \frac{1}{7}

\tan \frac{A}{2} = - \frac{\sqrt{2}}{10}

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detailed solution

Correct option is A

$\sin A = - 7 / 25$

$\Rightarrow \cos A = 24 / 25$ ( As C lies in fourth quadrant)

$\Rightarrow \sin 2 A = - \frac{336}{625}$

Now , $135^{\circ} < A / 2 < 180^{\circ}$

$2 \cos^{2} (A / 2) = 1 + \cos A = 49 / 25$

$\Rightarrow \cos^{2} (A / 2) = 49 / 50$

$\Rightarrow \cos (A / 2) = - \frac{7}{5 \sqrt{2}}$

$2 \sin^{2} (A / 2) = 1 - \cos A = 1 / 25$

$\Rightarrow \sin (A / 2) = \frac{1}{5 \sqrt{2}} \Rightarrow \tan (A / 2) = - \frac{1}{7}$

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If 3π2<A<2π then sin⁡A=−7/25 then,