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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2014 · Mathematics · Trigonometric Ratios & Identities

What is (1+ A)/(1- A)-(1- A)/(1+ A) equal to?

What is \frac{1+\sin A}{1-\sin A}-\frac{1-\sin A}{1+\sin A}$ equal to?

Options

  1. Asec⁡A−tan⁡A\sec \mathrm{A}-\tan \mathrm{A}
  2. B2sec⁡A⋅tan⁡A2 \sec \mathrm{ A} \cdot \tan \mathrm{ A}
  3. C4sec⁡A⋅tan⁡A4 \sec \mathrm{ A} \cdot \tan \mathrm{ A}
  4. D4cosec⁡A⋅cot⁡A4 \operatorname{cosec} A \cdot \cot A

Correct answer

C. 4sec⁡A⋅tan⁡A4 \sec \mathrm{ A} \cdot \tan \mathrm{ A}

Step-by-step solution

1+sin⁡A1−sin⁡A−1−sin⁡A1+sin⁡A=(1+sin⁡A)2−(1−sin⁡A)2(1−sin⁡A)(1+sin⁡A)=(1+2sin⁡A+sin⁡2A)−(1−2sin⁡A+sin⁡2A)1−sin⁡2A=1+2sin⁡A+sin⁡2A−1+2sin⁡A−sin⁡2Acos⁡2A=4sin⁡Acos⁡A×cos⁡A=4sec⁡A⋅tan⁡A.\begin{aligned} & \frac{1+\sin \mathrm{A}}{1-\sin \mathrm{A}}-\frac{1-\sin \mathrm{A}}{1+\sin \mathrm{A}} \\ = & \frac{(1+\sin \mathrm{A})^2-(1-\sin \mathrm{A})^2}{(1-\sin \mathrm{A})(1+\sin \mathrm{A})} \\ = & \frac{\left(1+2 \sin \mathrm{ A}+\sin ^2 \mathrm{ A}\right)-\left(1-2 \sin \mathrm{ A}+\sin ^2 \mathrm{ A}\right)}{1-\sin ^2 \mathrm{ A}} \\ = & \frac{1+2 \sin \mathrm{ A}+\sin ^2 \mathrm{ A}-1+2 \sin \mathrm{ A}-\sin ^2 \mathrm{ A}}{\cos ^2 \mathrm{ A}} \\ = & \frac{4 \sin \mathrm{ A}}{\cos \mathrm{ A} \times \cos \mathrm{A}}=4 \sec \mathrm{ A} \cdot \tan \mathrm{ A}.\end{aligned}
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