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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2019 · Mathematics · Trigonometric Ratios & Identities

What is A+cosec A equal to?

What is cot⁡A+cosec⁡A\cot \mathrm{A}+\operatorname{cosec} \mathrm{A} equal to?

Options

  1. Atan⁡(A2)\tan \left(\frac{A}{2}\right)
  2. Bcot⁡(A2)\cot \left(\frac{A}{2}\right)
  3. C2tan⁡(A2)2 \tan \left(\frac{A}{2}\right)
  4. D2cot⁡(A2)2 \cot \left(\frac{A}{2}\right)

Correct answer

B. cot⁡(A2)\cot \left(\frac{A}{2}\right)

Step-by-step solution

cot⁡A+cosec⁡A=cos⁡Asin⁡A+1sin⁡A=1+cos⁡Asin⁡A=2cos⁡2A22sin⁡A2⋅cos⁡A2=cot⁡(A2)\begin{aligned} & \cot A+\operatorname{cosec} A \\ & =\frac{\cos A}{\sin A}+\frac{1}{\sin A}=\frac{1+\cos A}{\sin A} \\ & =\frac{2 \cos ^2 \frac{A}{2}}{2 \sin \frac{A}{2} \cdot \cos \frac{A}{2}}=\cot \left(\frac{A}{2}\right)\end{aligned}$$
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