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

NDA 2014 · Mathematics · Trigonometric Ratios & Identities

What is ( 7 x- 3 x)/( 7 x-2 5 x+ 3 x) equal to?

What is \frac{\cos 7 x-\cos 3 x}{\sin 7 x-2 \sin 5 x+\sin 3 x}$ equal to?

Options

  1. Atan⁡x\tan x
  2. Bcot⁡x\cot x
  3. Ctan⁡2x\tan 2 x
  4. Dcot⁡2x\cot 2 x

Correct answer

B. cot⁡x\cot x

Step-by-step solution

cos⁡7x−cos⁡3xsin⁡7x+sin⁡3x−2sin⁡5x=−2sin⁡5x⋅sin⁡2x2sin⁡5x⋅cos⁡2x−2sin⁡5x=−2sin⁡5x⋅sin⁡2x−2sin⁡5x(1−cos⁡2x)=sin⁡2x2sin⁡2x=2sin⁡x⋅cos⁡x2sin⁡x⋅sin⁡x=cot⁡x.\begin{aligned} & \frac{\cos 7 x-\cos 3 x}{\sin 7 x+\sin 3 x-2 \sin 5 x} \\ & =\frac{-2 \sin 5 x \cdot \sin 2 x}{2 \sin 5 x \cdot \cos 2 x-2 \sin 5 x} \\ & =\frac{-2 \sin 5 x \cdot \sin 2 x}{-2 \sin 5 x(1-\cos 2 x)}=\frac{\sin 2 x}{2 \sin ^2 x} \\ & =\frac{2 \sin x \cdot \cos x}{2 \sin x \cdot \sin x}=\cot x.\end{aligned}
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