Question:
If $\alpha+\beta+\gamma=2\pi$, then $\tan\frac{\alpha}{2}+\tan\frac{\beta}{2}+\tan\frac{\gamma}{2} = $ ________.
If $\alpha+\beta+\gamma=2\pi$, then $\tan\frac{\alpha}{2}+\tan\frac{\beta}{2}+\tan\frac{\gamma}{2} = $ ________.
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Sum = Product for tangents of angles that sum to $\pi$.
Updated On: Jun 26, 2026
- $\tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$
- $\tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$ (Duplicate)
- $2\tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$
- $3\tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$
- $4\tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Apply the property of sum of tangents for half angles.
Step 2: Meaning
If $\frac{\alpha}{2} + \frac{\beta}{2} + \frac{\gamma}{2} = \pi$, then $\tan(\frac{\alpha}{2} + \frac{\beta}{2}) = \tan(\pi - \frac{\gamma}{2})$.
Step 3: Analysis
$\frac{\tan\frac{\alpha}{2} + \tan\frac{\beta}{2}}{1 - \tan\frac{\alpha}{2}\tan\frac{\beta}{2}} = -\tan\frac{\gamma}{2}$.
Step 4: Conclusion
Multiplying out gives $\tan\frac{\alpha}{2} + \tan\frac{\beta}{2} + \tan\frac{\gamma}{2} = \tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$. Final Answer: (A)
Apply the property of sum of tangents for half angles.
Step 2: Meaning
If $\frac{\alpha}{2} + \frac{\beta}{2} + \frac{\gamma}{2} = \pi$, then $\tan(\frac{\alpha}{2} + \frac{\beta}{2}) = \tan(\pi - \frac{\gamma}{2})$.
Step 3: Analysis
$\frac{\tan\frac{\alpha}{2} + \tan\frac{\beta}{2}}{1 - \tan\frac{\alpha}{2}\tan\frac{\beta}{2}} = -\tan\frac{\gamma}{2}$.
Step 4: Conclusion
Multiplying out gives $\tan\frac{\alpha}{2} + \tan\frac{\beta}{2} + \tan\frac{\gamma}{2} = \tan\frac{\alpha}{2} \tan\frac{\beta}{2} \tan\frac{\gamma}{2}$. Final Answer: (A)
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