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If $\sec \theta=1 \frac{1}{4}$, then $\tan \frac{\theta}{2}=$
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Verified Answer
The correct answer is:
$\frac{1}{3}$
Given that
$\begin{aligned}
\sec \theta & =\frac{5}{4} \\
& \sec \theta=\frac{1+\tan ^2(\theta / 2)}{1-\tan ^2(\theta / 2)} \Rightarrow \frac{5}{4}=\frac{1+\tan ^2(\theta / 2)}{1-\tan ^2(\theta / 2)} \\
\Rightarrow & 5-5 \tan ^2(\theta / 2)=4+4 \tan ^2(\theta / 2) \\
& \Rightarrow 9 \tan ^2(\theta / 2)=1 \Rightarrow \tan (\theta / 2)=\frac{1}{3} .
\end{aligned}$
$\begin{aligned}
\sec \theta & =\frac{5}{4} \\
& \sec \theta=\frac{1+\tan ^2(\theta / 2)}{1-\tan ^2(\theta / 2)} \Rightarrow \frac{5}{4}=\frac{1+\tan ^2(\theta / 2)}{1-\tan ^2(\theta / 2)} \\
\Rightarrow & 5-5 \tan ^2(\theta / 2)=4+4 \tan ^2(\theta / 2) \\
& \Rightarrow 9 \tan ^2(\theta / 2)=1 \Rightarrow \tan (\theta / 2)=\frac{1}{3} .
\end{aligned}$
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