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Q.
In $\Delta ABC$, the minimum value of $\frac{\sum \cot ^{2} \frac{ A }{2} \cdot \cot ^{2} \frac{ B }{2}}{\prod \cot ^{2} \frac{ A }{2}}$ is :
Trigonometric Functions
Solution:
$\frac{\sum \cot ^{2} \frac{ A }{2} \cdot \cot ^{2} \frac{ B }{2}}{\pi \cot ^{2} \frac{ A }{2}}=\tan ^{2} \frac{ A }{2}+\tan ^{2} \frac{ B }{2}+\tan ^{2} \frac{ C }{2}$
Now we have $\left(\tan \frac{ A }{2}-\tan \frac{ B }{2}\right)^{2} \geq 0$
$\tan ^{2} \frac{ A }{2}+\tan ^{2} \frac{ B }{2} \geq 2 \tan ^{2} \frac{ A }{2} \tan \frac{ B }{2}$
Similarly write this inequality for other angles and odd all in equalities, we get $\sum \tan \frac{ A }{2} \tan \frac{ B }{2}=1$