Which of the following quantities are rational? (a)`sin((11pi)/(12))sin((5pi)/(12))` (b) `cos e c((9pi)/(10))sec((4pi)/5)` (c)`sin^4(pi/8)+cos^4(pi/8)` (d) `(1+cos((2pi)/9))(1+cos((4pi)/9))(1+cos((8pi)/9))`

Options

A. `sin((11pi)/(12)) sin((5pi)/(12))`
B. `co sec((9pi)/(10))sec((4pi)/(5))`
C. `sin^(4)((pi)/(8))+cos^(4)((pi)/(8))`
D. `(1+cos((2pi)/(9)))(1+cos((4pi)/(9)))(1+cos((8pi)/(9)))`

Answer

A, B, C, D

Explanation

(1) `sin((11pi)/(12))sin((5pi)/(12))=sin((pi)/(12))cos((pi)/(12))`
`=(1)/(2)sin((pi)/(6))=(1)/(4)inQ`
(2) `co sec((9pi)/(10))sec((4pi)/(5))=-co sec((pi)/(10))sec((pi)/(5))`
`=(1)/(-sin18^(@)cos36^(@))`
`=(16)/((sqrt(5)-1)(sqrt(5)+1))`
=`-4inQ`
(3) `sin^(4)((pi)/(8))+cos^(4)((pi)/(8))=1-2sin^(2)((pi)/(8))cos^(2)((pi)/(8))`
`=1-(1)/(2)sin^(2)((pi)/(4))`
`=1-(1)/(4)=(3)/(4)inQ`
(4) `2cos^(2)((pi)/(9))2cos^(2)((2pi)/(9))2cos^(2)((4pi)/(9))`
`=8(cos20^(@)cos40cos80^(@))^(2)=(1)/(8)inQ`.