Question

cos 12° cos 24° cos 48° cos 96°​

Answer 1

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

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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