Find the value of the expression:-
3[sin^4 (3pi/2-x) + sin^4 (3pi+x)] -2 * [{sin^6 (pi/2+x) + sin^6 (5pi - x)]

Consider the following expression.    3 sin43π2-x+sin43π+x-2 sin6π2+x+sin65π-xRewrite the above expression as,   = 3 sin3π2-x4+sin3π+x4-2 sinπ2+x6+sin5π-x6 Note that,     sin3π2-x=sinπ2+π-x                             =cosπ-x                             =-cosx     sin3π+x=-sinx    3π+x  lies in 3rd quadrant and sine is negative      sinπ2+x=cosx           sin5π-x=sinx    5π-x  lies in 2nd quadrant and sine is positiveThis gives,     = 3-cosx4+-sinx4-2 cosx6+sinx6     = 3cos4x+sin4x-2 cos6x+sin6x     = 3cos4x+sin4x+2sin2x cos2x-2sin2x cos2x-2 cos2x3+sin2x3 Use the formulas,     a2+b2+2ab=a+b2             a3+b3=a+b a2+b2-abThis implies that,     = 3cos2x+sin2x2-2sin2x cos2x-2 cos2x+sin2x cos4x+sin4x-cos2x sin2x     = 31-2sin2x cos2x-2 1 cos4x+sin4x-cos2x sin2x       = 3-6sin2x cos2x-2cos4x+sin4x+2sin2x cos2x       = 3-2cos4x+sin4x-4sin2x cos2x      = 3-2cos4x+sin4x+2sin2x cos2x     = 3-2cos2x+sin2x2     = 3-21     = 3-2     =1    

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