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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
$\cot ^{-1}\left[(\cos \alpha)^{\frac{1}{2}}\right]-\tan ^{-1}\left[(\cos \alpha)^{\frac{1}{2}}\right]=x$, then $\sin x=$
  • ✓
    $\tan ^2\left(\frac{\alpha}{2}\right)$
  • B
    $\cot ^2\left(\frac{\alpha}{2}\right)$
  • C
    $\tan \alpha$
  • D
    $\cot \left(\frac{\alpha}{2}\right)$
✓

Answer

Correct option: A.
$\tan ^2\left(\frac{\alpha}{2}\right)$
(A) $\tan ^{-1}\left[\frac{1}{\sqrt{\cos \alpha}}\right]-\tan ^{-1}[\sqrt{\cos \alpha}]=x$
$\Rightarrow \tan ^{-1}\left[\frac{\frac{1}{\sqrt{\cos \alpha}}-\sqrt{\cos \alpha}}{1+\frac{\sqrt{\cos \alpha}}{\sqrt{\cos \alpha}}}\right]=x$
$\Rightarrow \tan x=\frac{1-\cos \alpha}{2 \sqrt{\cos \alpha}}$
$\therefore \quad \sin x=\frac{1-\cos \alpha}{1+\cos \alpha}=\frac{2 \sin ^2 \frac{\alpha}{2}}{2 \cos ^2 \frac{\alpha}{2}}=\tan ^2\left(\frac{\alpha}{2}\right)$

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