If =1 1 4 , then 2 = - Vidyadip

MCQ

If $\sec \theta=1 \frac{1}{4}$, then $\tan \frac{\theta}{2}=$

✓
$\frac{1}{3}$
B
$\frac{3}{4}$
C
$\frac{1}{4}$
D
$\frac{5}{4}$

✓

Answer

Correct option: A.

$\frac{1}{3}$

(A)
Given that, $\sec \theta=\frac{5}{4}$
Since, $\sec \theta=\frac{1+\tan ^2\left(\frac{\theta}{2}\right)}{1-\tan ^2\left(\frac{\theta}{2}\right)}$
$\Rightarrow \frac{5}{4}=\frac{1+\tan ^2\left(\frac{\theta}{2}\right)}{1-\tan ^2\left(\frac{\theta}{2}\right)}$
$\Rightarrow 5-5 \tan ^2\left(\frac{\theta}{2}\right)=4+4 \tan ^2\left(\frac{\theta}{2}\right)$
$\Rightarrow 9 \tan ^2\left(\frac{\theta}{2}\right)=1$
$\Rightarrow \tan \left(\frac{\theta}{2}\right)=\frac{1}{3}$

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