something is wrong here... on the left side you have sin( A ) and on the right side x ???

Anyway... if A = x...

sin^6( x ) + cos^6( x ) = ( sin^2( x ) + cos^2( x ) )^3 - 3sin^4( x )*cos^2( x ) - 3sin^2( x )*cos^4( x ) =

= 1^3 - 3*( sin^2( 2x )/4 )*sin^2( x ) - 3( sin^2( 2x )/4 )*cos^2( x ) =

= 1 - (3/4)*sin^2( 2x )*( sin^2( x ) + cos^2( x ) ) =

= 1 - (3/4)*sin^2( 2x )