Question

If $\sec \theta = 1 \frac{1}{4}$, then $\tan \frac{\theta}{2} =$ A. $\frac{1}{3}$ B. $\frac{3}{4}$ C. $\frac{1}{4}$

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Answer

The correct option is A.

Given that $\sec \theta = \frac{5}{4}$:

$$ \sec \theta = \frac{1 + \tan^2\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} $$

Equate the given $\sec \theta$ to the above expression:

$$ \frac{5}{4} = \frac{1 + \tan^2\left(\frac{\theta}{2}\right)}{1 - \tan^2\left(\frac{\theta}{2}\right)} $$

Cross-multiplying to solve for $\tan^2\left(\frac{\theta}{2}\right)$:

$$ 5 - 5 \tan^2\left(\frac{\theta}{2}\right) = 4 + 4 \tan^2\left(\frac{\theta}{2}\right) $$

Combine like terms:

$$ 9 \tan^2\left(\frac{\theta}{2}\right) = 1 $$

Solving for $\tan \left(\frac{\theta}{2}\right)$:

$$ \tan \left(\frac{\theta}{2}\right) = \frac{1}{3} $$

Therefore, the value of $\tan \frac{\theta}{2}$ is

$$ \boxed{\frac{1}{3}} $$


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