1) cos( -x ) = cos( x ) // cos( x ) is even function

f( x ) = cos^2( x )

f( -x ) = cos^2( -x ) = cos( -x )*cos( -x ) = cos( x )*cos( x ) =

= cos^2( x ) = f( x )

f( x ) is even function

2) you can find formulas on internet for this kind of problems

(hint: cos( 4x ) = cos( 2x + 2x ) , sin( 3x ) = sin( 2x + x ) )

3)

sin( 2x ) = 2sin( x )*cos( x )

cos( x ) = sqrt( 1 - sin^2( x ) )

sin( 2*sin^-1( 4/5 ) ) = 2*sin( sin^-1( 4/5 ) )*cos( sin^-1( 4/5) ) = 2*( 4/5 )*sqrt( 1 - sin^2( sin^-1( 4/5 ) ) ) =

= ( 8/5 )*sqrt( 1 - 16/25 ) = ( 8/5 )*sqrt( 9/25 ) = ( 8/5 )*( 3/5 ) = 24/25