The value of `[ ( sin ""(pi)/(9)) (4+ sec""(pi)/(9))]^(2)` is _______.

Answer

3

Explanation

`(sin ""(pi)/(9)) (4+sec""(pi)/(9))`
` " " = sin 20^(@) ( 4+ (1)/( cos 20^(@)))`
` " " = ( 4 sin 20^(@) cos 20^(@) + sin 20^(@))/( cos 20^(@))`
`" " = ( 2 sin 40^(@) + sin 20^(@))/( cos 20^(@))`
`" " = (sin 40^(@) + ( sin 40^(@) + sin 20^(@))/(cos 20^(@))`
` ( sin 40^(@) + 2 sin 30^(@) cos 10^(@))/(cos 20^(@))`
`= ( sin 40^(@) + cos 10^(@))/( cos 20^(@))`
`= (cos 50^(@) + cos 10^(@))/(cos 20^(@))`
`= ( 2 cos 30^(@) cos 20^(@))/(cos 20^(@))`
` = sqrt3`