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sin(2x)  =  24/25

So

2sinxcosx =  24/25          square both sides

4sin^2(x)cos^2(x )  =  576/625

2sin^2(x)cos^2(x ) =  576/ [ 625 * 2 ]  =   288/625

cos(2x)   =   sqrt  ( 1 -  sin^2(2x) )   =  sqrt  ( 1 - 576/625)  =  sqrt ( 625 - 576] / 25  = sqrt (49) / 25 = 7/25

cos (2x)  =   cos^2x  - sin^2x

7/25  =  cos^2x - sin^2 x               square both sides

49/625  = cos^4x  -  2cos^2(x)sin^2(x) + sin^4x

49/625  =  sin^4x  +  cos^4x  -  288/625

[ 49  + 288 ] / 625   =  sin^4x + cos^4x

337/625  =  sin^4x + cos^4x