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INVERSE TRIGNOMETRIC FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINVERSE TRIGNOMETRIC FUNCTIONS1 Mark
Question
$\cot\Big(\frac{\pi}{4}-2\cot^{-1}3\Big)=$
  1. 7
  2. 6
  3. 5
  4. none of these
✓

Answer

  1. 7

Solution:

Let $2\cot^{-1}3=\text{y}$

Then, $\cot\frac{\text{y}}{2}=3$

$\cot\Big(\frac{\pi}{4}-2\cot^{-1}3\Big)=\cot\Big(\frac{\pi}{4}-\text{y}\Big)$

$=\frac{\cot\frac{\pi}{4}\cot\text{y}+1}{\cot\text{y}-\cot\frac{\pi}{4}}$

$=\frac{\cot\text{y}+1}{\cot\text{y}-1}$

$=\frac{\frac{\cot^2\frac{\text{y}}{2}-1}{2\cot\frac{\text{y}}{2}}+1}{\frac{\cot^2\frac{\text{y}}{2}-1}{2\cot\frac{\text{y}}{2}}-1}$

$=\frac{\cot^2\frac{\text{y}}{2}+2\cot\frac{\text{y}}{2}-1}{\cot^2\frac{\text{y}}{2}-2\cot\frac{\text{y}}{2}-1}$

$=\frac{9+6-1}{9-6-1}$

$=7$

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