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If $2 \cos \frac{A}{2} = \sqrt{1 + \sin A} + \sqrt{1 - \sin A},$ then $\frac{A}{2}$

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a

2nπ+π4 and 2nπ+3π4

b

2nπ−π4 and 2nπ+π4

c

2nπ−3π4 and 2nπ−π4

d

−∞ and +∞

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detailed solution

Correct option is B

We have,2cos⁡A2=1+sin⁡A+1−sin⁡A⇒2cos⁡A2=cos⁡A2+sin⁡A22+cos⁡A2−sin⁡A22⇒2cos⁡A2=cos⁡A2+sin⁡A2+cos⁡A2−sin⁡A2⇒cos⁡A2+sin⁡A2≥0 and cos⁡A2−sin⁡A2≥0⇒−3π4≤A2≤3π4 and −π4≤A2≤π4⇒−π4≤A2≤π4⇒ 2nπ−π4≤A2≤2nπ+π4,n∈Z

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