We can use the trigonometric identity sin ( A + B ) = sin ( A ) cos ( B ) + cos ( A ) sin ( B ) \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) sin ( A + B ) = sin ( A ) cos ( B ) + cos ( A ) sin ( B ) sin ( 5 π 12 ) sin ( π 12 ) = sin ( π 4 + π 6 ) sin ( π 12 ) \sin\left(\frac{5\pi}{12}\right) \sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) \sin\left(\frac{\pi}{12}\right) sin ( 12 5 π ) sin ( 12 π ) = sin ( 4 π + 6 π ) sin ( 12 π ) Using the identity, we have: sin ( π 4 + π 6 ) sin ( π 12 ) = ( sin ( π 4 ) cos ( π 6 ) + cos ( π 4 ) sin ( π 6 ) ) sin ( π 12 ) \sin\left(\frac{\pi}{4} + \frac{\pi}{6}\right) \sin\left(\frac{\pi}{12}\right) = \left(\sin\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right)\right) \sin\left(\frac{\pi}{12}\right) sin ( 4 π + 6 π ) sin ( 12 π ) = ( sin ( 4 π ) cos ( 6 π ) + cos ( 4 π ) sin ( 6 π ) ) sin ( 12 π )
Simplifying further: ( sin ( π 4 ) cos ( π 6 ) + cos ( π 4 ) sin ( π 6 ) ) sin ( π 12 ) = ( 1 2 ⋅ 3 2 + 1 2 ⋅ 1 2 ) sin ( π 12 ) (\sin(\frac{\pi}{4})\cos(\frac{\pi}{6}) + \cos(\frac{\pi}{4})\sin(\frac{\pi}{6}))\sin(\frac{\pi}{12}) = (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2})\sin(\frac{\pi}{12}) ( sin ( 4 π ) cos ( 6 π ) + cos ( 4 π ) sin ( 6 π )) sin ( 12 π ) = ( 2 1 ⋅ 2 3 + 2 1 ⋅ 2 1 ) sin ( 12 π ) Combining terms:( 1 2 ⋅ 3 2 + 1 2 ⋅ 1 2 ) sin ( π 12 ) = ( 3 + 1 2 ) ⋅ sin ( π 12 ) \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right)\sin\left(\frac{\pi}{12}\right) = \left(\frac{\sqrt{3} + 1}{2}\right) \cdot \sin\left(\frac{\pi}{12}\right) ( 2 1 ⋅ 2 3 + 2 1 ⋅ 2 1 ) sin ( 12 π ) = ( 2 3 + 1 ) ⋅ sin ( 12 π )
Now, sin ( π 12 ) \sin\left(\frac{\pi}{12}\right) sin ( 12 π ) can be simplified using the half-angle formula: sin ( π 12 ) = 1 − cos ( π 6 ) 2 = 1 − 3 2 2 \sin\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 - \cos\left(\frac{\pi}{6}\right)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} sin ( 12 π ) = 2 1 − c o s ( 6 π ) = 2 1 − 2 3
Substituting back into the expression: ( 3 + 1 2 ) sin ( π 12 ) = ( 3 + 1 2 ) 1 − 3 2 2 \left(\frac{\sqrt{3} + 1}{2}\right) \sin\left(\frac{\pi}{12}\right) = \left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} ( 2 3 + 1 ) sin ( 12 π ) = ( 2 3 + 1 ) 2 1 − 2 3 Simplifying the expression: ( 3 + 1 2 ) 1 − 3 2 2 = ( 3 + 1 2 ) 2 − 3 4 = ( 3 + 1 2 ) ( 2 − 3 2 ) \left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \left(\frac{\sqrt{3} + 1}{2}\right) \sqrt{\frac{2 - \sqrt{3}}{4}} = \left(\frac{\sqrt{3} + 1}{2}\right) \left(\frac{\sqrt{2} - \sqrt{3}}{2}\right) ( 2 3 + 1 ) 2 1 − 2 3 = ( 2 3 + 1 ) 4 2 − 3 = ( 2 3 + 1 ) ( 2 2 − 3 )
Multiplying the fractions:( 3 + 1 2 ) ( 2 − 3 2 ) = 6 − 3 + 2 − 1 4 \left(\frac{\sqrt{3} + 1}{2}\right) \left(\frac{\sqrt{2} - \sqrt{3}}{2}\right) = \frac{\sqrt{6} - \sqrt{3} + \sqrt{2} - 1}{4} ( 2 3 + 1 ) ( 2 2 − 3 ) = 4 6 − 3 + 2 − 1 Therefore, the value of sin ( 5 π 12 ) sin ( π 12 ) = 6 − 3 + 2 − 1 4 \sin\left(\frac{5\pi}{12}\right) \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{3} + \sqrt{2} - 1}{4} sin ( 12 5 π ) sin ( 12 π ) = 4 6 − 3 + 2 − 1 Hence, the correct option is (D) 1 4 \frac{1}{4} 4 1 .