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Question

Simplify:
$$\sin { 36 } .\sin { 27 }. \sin { 108 }. \sin { 144 } =\dfrac { 5 }{ 16 }$$.


Solution

$$\sin 36°\sin 72°\sin 108°\sin 144°$$

$$=\sin 36°\sin 72°\sin (180°-72°)\sin (180°-36°)$$

$$=\sin 36°\sin 72°\sin { (90°\ast 2)-72° }\sin { (90°\ast 2)-36° }$$ 

$$=\sin 36°\sin 72°\sin 72°\sin 36°$$ 

$$=\sin ^2 36°\sin^2 72°$$

$$={ \left[ \dfrac { \sqrt { 10 } -2\sqrt { 5 }  }{ 4 }  \right]  }^{ 2 }\ast { \left[ \dfrac { \sqrt { 10 } +2\sqrt { 5 }  }{ 4 }  \right]  }^{ 2 }$$ 

$$={ \left[ \dfrac { \sqrt { 10 } -2\sqrt { 5 }  }{ 16 }  \right]  }\ast { \left[ \dfrac { \sqrt { 10 } +2\sqrt { 5 }  }{ 16 }  \right]  }$$

$$=\dfrac { (100-20) }{ (16×16) } $$

$$=\dfrac { 80 }{ (16×16) } $$

$$=\dfrac { 5 }{ 16 } $$(Proved)

Mathematics

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