Question

cos 31°cos1°-sin46°sin14°​

Answer 1

Step-by-step explanation:

$31x1 = 46x14 \\ 31 = 644$

Answer 2

SOLUTION

TO DETERMINE

The value of

$\sf cos \: {31}^{ \circ} cos \: {1}^{ \circ} - sin \: {46}^{ \circ} sin \: {14}^{ \circ}$

EVALUATION

Here the given expression is

$\sf cos \: {31}^{ \circ} cos \: {1}^{ \circ} - sin \: {46}^{ \circ} sin \: {14}^{ \circ}$

We calculate the value as below

$\sf cos \: {31}^{ \circ} cos \: {1}^{ \circ} - sin \: {46}^{ \circ} sin \: {14}^{ \circ}$

$\displaystyle \sf{ = \frac{1}{2} \bigg(2 cos \: {31}^{ \circ} cos \: {1}^{ \circ} - 2sin \: {46}^{ \circ} sin \: {14}^{ \circ} } \bigg)$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ cos \:( {31}^{ \circ} + {1}^{ \circ} ) +cos \:( {31}^{ \circ} - {1}^{ \circ}) \bigg] -\frac{1}{2} \bigg[ cos \:( {46}^{ \circ} - {14}^{ \circ} ) - cos \:( {46}^{ \circ} + {14}^{ \circ}) \bigg] }$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ cos \: {32}^{ \circ} +cos \: {30}^{ \circ} \bigg] -\frac{1}{2} \bigg[ cos \: {32}^{ \circ} - cos \ {60}^{ \circ} \bigg] }$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ cos \: {32}^{ \circ} +cos \: {30}^{ \circ} - cos \: {32}^{ \circ} + cos \ {60}^{ \circ} \bigg] }$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ cos \: {30}^{ \circ} + cos \ {60}^{ \circ} \bigg] }$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ \frac{ \sqrt{3} }{2} + \frac{1}{2} \bigg] }$

$\displaystyle \sf{ = \frac{1}{2} \bigg[ \frac{ \sqrt{3} + 1 }{2} \bigg] }$

$\displaystyle \sf{ = \frac{ \sqrt{3} + 1 }{4} }$

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