Question
If $$\left( {\sec \,\alpha + \tan \,\alpha } \right)\left( {\sec \,\beta + \tan \,\beta } \right)\left( {\sec \,\gamma + \tan \,\gamma } \right) = \,\tan \,\alpha \,\tan \,\beta \,\tan \,\gamma ,$$ then expression $$\left( {\sec \,\alpha - \tan \,\alpha } \right)\left( {\sec \,\beta - \tan \,\beta } \right)\left( {\sec \,\gamma - \tan \,\gamma } \right)$$ is equal to
A.
$$\cot \alpha \cot \beta \cot \gamma $$
B.
$$\tan \alpha \tan \beta \tan \gamma $$
C.
$$\cot \alpha + \cot \beta + \cot \gamma $$
D.
$$\tan \alpha + \tan \beta + \tan \gamma $$
Answer :
$$\cot \alpha \cot \beta \cot \gamma $$
Solution :
$$\eqalign{
& \left( {\sec \,\alpha + \tan \,\alpha } \right)\left( {\sec \,\beta + \tan \,\beta } \right)\left( {\sec \,\gamma + \tan \,\gamma } \right) = \,\tan \,\alpha \,\tan \,\beta \,\tan \,\gamma \cr
& \Rightarrow \,\left( {{{\sec }^2}\,\alpha - {{\tan }^2}\alpha } \right)\left( {{{\sec }^2}\beta - {{\tan }^2}\beta } \right)\left( {{{\sec }^2}\gamma - {{\tan }^2}\gamma } \right) \cr
& = \tan \alpha \,\tan\beta \,\tan \gamma \left( {\sec \alpha - \tan \alpha } \right)\left( {\sec \beta - \tan\beta } \right)\left( {\sec \gamma - \tan \gamma } \right) \cr
& \Rightarrow \,\left( {\sec \alpha - \tan \alpha } \right)\left( {\sec \beta - \tan\beta } \right)\left( {\sec \gamma - \tan \gamma } \right) \cr
& = \cot \alpha \cot \beta \,\cot \gamma \cr} $$