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Trigonometry – I — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – I1 Mark
MCQ
The value of $\tan 1^{\circ} \cdot \tan 2^{\circ} \tan 3^{\circ}$ equal to
  • A
    -1
  • ✓
    1
  • C
    $\frac{\pi}{2}$
  • D
    2
✓

Answer

Correct option: B.
1
1
Explanation:
$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$
$=\left(\tan 1^{\circ} \tan 89^{\circ}\right)\left(\tan 2^{\circ} \tan 88^{\circ}\right)$
$\ldots\left(\tan 44^{\circ} \tan 46^{\circ}\right) \tan 45^{\circ}$
$=\left(\tan 1^{\circ} \cot 1^{\circ}\right)\left(\tan 2^{\circ} \cot 2^{\circ}\right)$
$\ldots\left(\tan 44^{\circ} \cot 44^{\circ}\right) \cdot \tan 45^{\circ}$
$\left.\ldots \tan \left(\because 90^{\circ}-\theta\right)=\cot \theta\right]$
$=1 \times 1 \times 1 \times \ldots \times 1 \times \tan 45^{\circ}=1$
$\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ} \ldots \tan 89^{\circ}$
$=\left(\tan 1^{\circ} \tan 89^{\circ}\right)\left(\tan 2^{\circ} \tan 88^{\circ}\right)$
$\ldots\left(\tan 44^{\circ} \tan 46^{\circ}\right) \tan 45^{\circ}$
$=\left(\tan 1^{\circ} \cot 1^{\circ}\right)\left(\tan 2^{\circ} \cot 2^{\circ}\right)$
$\ldots\left(\tan 44^{\circ} \cot 44^{\circ}\right) \cdot \tan 45^{\circ}$
$\left.\ldots \tan \left(\because 90^{\circ}-\theta\right)=\cot \theta\right]$
$=1 \times 1 \times 1 \times \ldots \times 1 \times \tan 45^{\circ}=1$

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