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Mathematics

(1−sin⁡A1+sin⁡A)\sqrt{\Big(\dfrac{1 − \sin A}{1 + \sin A}\Big)} = ?

  1. sec A + tan A

  2. sec A − tan A

  3. sec A tan A

  4. none of these

Trigonometric Identities

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Answer

Rationalize the expression,

⇒(1−sin⁡A1+sin⁡A)⇒(1−sin⁡A1+sin⁡A)×1−sin⁡A1−sin⁡A⇒((1−sin⁡A)21−sin⁡2A)⇒((1−sin⁡A)2cos⁡2A)⇒(1−sin⁡A)cos⁡A⇒1cos⁡A−sin⁡Acos⁡A⇒sec⁡A−tan⁡A\Rightarrow \sqrt{\Big(\dfrac{1 − \sin A}{1 + \sin A}\Big)} \\[1em] \Rightarrow \sqrt{\Big(\dfrac{1 − \sin A}{1 + \sin A}\Big) \times \dfrac{1 - \sin A}{1 - \sin A}} \\[1em] \Rightarrow \sqrt{\Big(\dfrac{(1 − \sin A)^2}{1 - \sin^2 A}\Big)} \\[1em] \Rightarrow \sqrt{\Big(\dfrac{(1 − \sin A)^2}{\cos^2 A}\Big)} \\[1em] \Rightarrow \dfrac{(1 − \sin A)}{\cos A} \\[1em] \Rightarrow \dfrac{1}{\cos A} − \dfrac{\sin A}{\cos A} \\[1em] \Rightarrow \sec A − \tan A

Hence, option 2 is the correct option.

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