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 :
Given: \[(\sec \alpha +\tan \alpha )(\sec \beta +\tan \beta )(\sec \gamma +\tan \gamma )\] \[=\tan \alpha \tan \beta \tan \gamma \] ...(i) Let\[x=(\sec \alpha -\tan \alpha )(\sec \beta -\tan \beta )(\sec \gamma -\tan \gamma )\] ...(ii) Multiply both equations, (i) and (ii), we get \[({{\sec }^{2}}\alpha -{{\tan }^{2}}\alpha )({{\sec }^{2}}\beta -{{\tan }^{2}}\beta )({{\sec }^{2}}\gamma -{{\tan }^{2}}\gamma )\] \[=x.(\tan \alpha \tan \beta \tan \gamma )\] \[\Rightarrow x=\frac{1}{\tan \alpha \tan \beta \tan \gamma }\] \[\therefore x=\cot \alpha \cot \beta \cot \gamma \]
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