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Son 12'sin 48° sin s4 =​

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Answer:

 Let  x = sin54 SIN48 sin12  

          = (1/2) x sin 54 x ( 2 sin48 sin 12 )

We KNOW, 2 sin A sin B =cos(A − B) − cos(A + B)

          = 1/2 x sin54 x (cos36-cos60) 

We know, sin( 90 - x ) = cos x and cos 60 = 1/2

          =1/2 x cos36 x (cos36-1/2) 

          =1/2 x (cos36)² - 1/4 x cos36

We know, cos² x = 1 + cos 2x  

          =1/4 x (1+cos72) - 1/4 x cos36 

          =1/4 + (1/4)(cos72-cos36)

We know, cos A − cos B = - 2 sin (A + B/ 2)  sin (A − B/ 2) 

 

          =1/4-1/2 x sin54 x sin18 

     We know, sin 90-x  = cos x

          =1/4-1/2 x cos36 x sin18 

We MULTIPLY both sides with 8cos18 

8cos18x = 8cos18 x (1/4-1/2 x cos36 x sin18) 

8cos18x = 2cos18 - 4cos36 x sin18 x cos18 

We know, sin 2 x =2 sin x cos x

 

8cos18x = 2cos18 - 2cos36 x sin36 

8cos18x= 2cos18-sin72 

8cos18x= 2cos18-cos18 

8cos18x = cos18

Thus, 

 x=cos18/(8cos18)

  =1/8


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