A=580∘⇒A2=290∘ ∴ 2sin⁡A2=−1+sin⁡A−1−sin⁡A and 2cos⁡A2=−1+sin⁡A+1−sin⁡A

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If A = 580^o, then

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$\sin \frac{1}{2} A = \frac{1}{2} [\sqrt{(1 + \sin A)} - \sqrt{(1 - \sin A)}]$

$\sin \frac{1}{2} A = - \sqrt{(1 + \sin A)} - \sqrt{(1 - \sin A)}$

$\sin \frac{1}{2} A = \frac{1}{2} [- \sqrt{(1 + \sin A)} - \sqrt{(1 - \sin A)}]$

$\cos \frac{1}{2} A = \sqrt{(1 + \sin A)} - \sqrt{(1 - \cos A)}$

answer is C.

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Detailed Solution

$A = 580^{\circ} \Rightarrow \frac{A}{2} = 290^{\circ}$

$\begin{aligned} ∴ 2 \sin \frac{A}{2} = - \sqrt{1 + \sin A} - \sqrt{1 - \sin A} \\ and 2 \cos \frac{A}{2} = - \sqrt{1 + \sin A} + \sqrt{1 - \sin A} \end{aligned}$

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