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))`

Explanation

a)`sin(11/12pi)sin(5/12pi)=sin(pi-pi/12)cos(pi/2-5/12pi)`

`=sinpi/12*cospi/12]*2/2`

`=1/2*sinpi/6=1/2*1/2=1/4`

b)`cosec(9/10pi)sec(4/5pi)=cosec(pi-pi/10)sec(pi-pi/5)`

`=cosecpi/10*(-secpi/5)`

`=-1/(sin18*cos36)`

`=-1/(sqrt5-1)(sqrt5+1)`

`=-16/4=-4`.

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