Assuming you mean to solve the equation ( \sin^4(x) + \cos^4(x) - \cos(2x) = 0.5 ), we can simplify it as follows:

Since ( \cos(2x) = 2\cos^2(x) - 1 ), we can substitute this into the equation:

( \sin^4(x) + \cos^4(x) - (2\cos^2(x) - 1) = 0.5 )

Expand ( \sin^4(x) ) and ( \cos^4(x) ) into squares:

( (\sin^2(x) + \cos^2(x))^2 - 2\cos^2(x) + 1 = 0.5 )

Since ( \sin^2(x) + \cos^2(x) = 1 ), the equation becomes:

( 1 - 2\cos^2(x) + 1 = 0.5 )

( -2\cos^2(x) + 2 = 0.5 )

( -2\cos^2(x) = -1.5 )

( \cos^2(x) = 0.75 )

( \cos(x) = \pm \sqrt{0.75} )

Therefore, the solutions for ( x ) are:

( x = \cos^{-1}(\sqrt{0.75}) ) and ( x = \cos^{-1}(-\sqrt{0.75}) )

These values can be further computed for exact numerical values.