Math, asked by ayushsrivas55, 11 months ago

cos 12° cos 24° cos 48° cos 96°​

Answers

Answered by pulakmath007
4

\displaystyle \sf{cos  \:  {12}^{ \circ}  \:cos  \:  {24}^{ \circ}  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    =  -  \frac{1}{16}  }

Given :

\displaystyle \sf{cos  \:  {12}^{ \circ}  \:cos  \:  {24}^{ \circ}  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

To find :

The value of the expression

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

cos 12° cos 24° cos 48° cos 96°

Step 2 of 2 :

Find the value of the expression

\displaystyle \sf{cos  \:  {12}^{ \circ}  \:cos  \:  {24}^{ \circ}  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{2 \: sin \:  {12}^{ \circ}}(2\: sin \:  {12}^{ \circ} cos  \:  {12}^{ \circ} ) \:cos  \:  {24}^{ \circ}  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{2 \: sin \:  {12}^{ \circ}}\: sin \:  {24}^{ \circ}  \:cos  \:  {24}^{ \circ}  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{2}  \: sin \:  {12}^{ \circ}}\: (2sin \:  {24}^{ \circ}  \:cos  \:  {24}^{ \circ})  \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{2}  \: sin \:  {12}^{ \circ}}\: sin \:  {48}^{ \circ}    \: cos  \:  {48}^{ \circ}  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{3}  \: sin \:  {12}^{ \circ}}\: (2sin \:  {48}^{ \circ}    \: cos  \:  {48}^{ \circ})  \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{3}  \: sin \:  {12}^{ \circ}}\: sin \:  {96}^{ \circ}    \: cos  \:  {96}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{4}  \: sin \:  {12}^{ \circ}}\: (2sin \:  {96}^{ \circ}    \: cos  \:  {96}^{ \circ} )   }

\displaystyle \sf{ =  \frac{1}{ {2}^{4}  \: sin \:  {12}^{ \circ}}\: sin \:  {192}^{ \circ}    }

\displaystyle \sf{ =  \frac{1}{ {2}^{4}  \: sin \:  {12}^{ \circ}}\: sin \:  ({180}^{ \circ} + {12}^{ \circ} )   }

\displaystyle \sf{ =  \frac{1}{ {2}^{4}  \: sin \:  {12}^{ \circ}}\:( -  sin  \:  {12}^{ \circ} )   }

\displaystyle \sf{ =   \frac{ - 1}{ {2}^{4} }  }

\displaystyle \sf{ =    -  \frac{1}{16}  }

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