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INVERSE TRIGNOMETRIC FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan 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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