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BITSAT 2016
Exam
BITSAT
Year
2016
Subject
Mathematics
Chapter
Trigonometric Equations

BITSAT 2016 · Mathematics · Trigonometric Equations

If ( x)= ( x) , then 2 x is equal to:

If tan⁡(cot⁡x)=cot⁡(tan⁡x)\tan (\cot x)=\cot (\tan x), then sin⁡2x\sin 2 x is equal to:

Options

  1. A2(2n+1)π\frac{2}{(2 n+1) \pi}
  2. B4(2n+1)π\frac{4}{(2 n+1) \pi}
  3. C2n(n+1)π\frac{2}{n(n+1) \pi}
  4. D4n(n+1)π\frac{4}{n(n+1) \pi}

Correct answer

B. 4(2n+1)π\frac{4}{(2 n+1) \pi}

Step-by-step solution

Given, tan⁡(cot⁡x)=cot⁡(tan⁡x)=tan⁡(π2−tan⁡x)⇒cot⁡x=nπ+π2−tan⁡x⇒cot⁡x+tan⁡x=nπ+π2⇒1sin⁡xcos⁡x=nπ+π2⇒1sin⁡2x=nπ2+π4⇒sin⁡2x=1nπ2+π4=4(2n+1)π\tan (\cot x)=\cot (\tan x)=\tan \left(\frac{\pi}{2}-\tan x\right)\Rightarrow \cot x=n \pi+\frac{\pi}{2}-\tan x\Rightarrow \cot x+\tan x=n \pi+\frac{\pi}{2}\Rightarrow \frac{1}{\sin x \cos x}=n \pi+\frac{\pi}{2} \Rightarrow \frac{1}{\sin 2 x}=\frac{n \pi}{2}+\frac{\pi}{4}\Rightarrow \sin 2 x=\frac{1}{\frac{n \pi}{2}+\frac{\pi}{4}}=\frac{4}{(2 n+1) \pi}
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