Which of the following quantities are rational?

sin (\frac{11 π}{12}) sin (\frac{5 π}{12})

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cose c (\frac{9 π}{10}) sec (\frac{4 π}{5})

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{sin}^{4} (\frac{π}{8}) + {cos}^{4} (\frac{π}{8})

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(1 + cos \frac{2 π}{9}) (1 + cos \frac{4 π}{9}) (1 + cos \frac{8 π}{9})

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Solution

The correct options are
A $sin (\frac{11 π}{12}) sin (\frac{5 π}{12})$
B $cose c (\frac{9 π}{10}) sec (\frac{4 π}{5})$
C ${sin}^{4} (\frac{π}{8}) + {cos}^{4} (\frac{π}{8})$
D $(1 + cos \frac{2 π}{9}) (1 + cos \frac{4 π}{9}) (1 + cos \frac{8 π}{9})$
$a = sin \frac{11 π}{12} sin \frac{5 π}{12} \Rightarrow a = \frac{1}{2} (cos (\frac{11 π - 5 π}{12}) - cos (\frac{11 π + 5 π}{12})) \Rightarrow a = \frac{1}{2} (cos \frac{π}{2} - cos \frac{4 π}{3}) = \frac{1}{4}$
$b = csc \frac{9 π}{10} sec \frac{4 π}{5} \Rightarrow b = \frac{1}{sin \frac{9 π}{10} cos \frac{4 π}{5}} = \frac{1}{sin (π - \frac{π}{10}) cos (π - \frac{π}{5})} \Rightarrow b = - \frac{2 cos \frac{π}{10}}{(2 cos \frac{π}{10} sin \frac{π}{10}) cos \frac{π}{5}} \Rightarrow b = - \frac{4 cos (\frac{π}{2} - \frac{2 π}{5})}{2 sin \frac{π}{5} cos \frac{π}{5}} \Rightarrow b = - \frac{4 sin \frac{2 π}{5}}{sin \frac{2 π}{5}} = - 4$
$c = {sin}^{4} \frac{π}{8} + {cos}^{4} \frac{π}{8} \Rightarrow c = {⎛ ⎜ ⎜ ⎝ \frac{1 - cos \frac{π}{4}}{2} ⎞ ⎟ ⎟ ⎠}^{2} + {⎛ ⎜ ⎜ ⎝ \frac{1 + cos \frac{π}{4}}{2} ⎞ ⎟ ⎟ ⎠}^{2} \Rightarrow c = 1 + {cos}^{2} \frac{π}{4} = \frac{3}{2}$
$d = (1 + cos \frac{2 π}{9}) (1 + cos \frac{4 π}{9}) (1 + cos \frac{8 π}{9}) \Rightarrow d = 8 {cos}^{2} \frac{π}{9} {cos}^{2} \frac{2 π}{9} {cos}^{2} \frac{4 π}{9} \Rightarrow d = 8 {⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{(2 cos \frac{π}{9} sin \frac{π}{9}) (2 cos \frac{2 π}{9} sin \frac{2 π}{9}) (2 cos \frac{4 π}{9} sin \frac{4 π}{9})}{8 sin \frac{π}{9} sin \frac{2 π}{9} sin \frac{4 π}{9}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}^{2} \Rightarrow d = 8 {⎛ ⎜ ⎜ ⎜ ⎝ \frac{sin \frac{2 π}{9} sin \frac{4 π}{9} sin \frac{2 π}{9}}{8 sin \frac{π}{9} sin \frac{2 π}{9} sin \frac{4 π}{9}} ⎞ ⎟ ⎟ ⎟ ⎠}^{2} = \frac{1}{8}$
Ans: A,B,C,D

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