A) \[\cot \alpha \,\,\cot \beta \,\,\cot \gamma \]
B) \[\tan \alpha \,\,tan\beta \,\,tan\gamma \]
C) \[\cot \alpha +\cot \beta +\cot \gamma \]
D) \[tan\alpha +tan\beta +tan\gamma \]
Correct Answer: A
Solution :
\[(\sec \alpha +\tan \alpha )(\sec \beta +\tan \beta )(\sec \gamma +\tan \gamma )\] \[=\tan \alpha \tan \beta \tan \gamma \] \[\Rightarrow \,({{\sec }^{2}}\alpha -{{\tan }^{2}}\alpha )({{\sec }^{2}}\beta -{{\tan }^{2}}\beta )({{\sec }^{2}}\gamma -{{\tan }^{2}}\gamma )\]\[=\tan \alpha tan\beta tan\gamma (sec\alpha -tan\alpha )(sec\beta -tan\beta )\] \[(\sec \gamma -\tan \gamma )\] \[\Rightarrow \,\,(\sec \alpha -\tan \alpha )(\sec \beta -\tan \beta )(\sec \gamma -\tan \gamma )\] \[=\cot \alpha \,\,\cot \beta \,\,\cot \gamma \]
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