1.$\displaystyle (1-2sin^2(x))^2 + 4sin^2(x) cos^2(x) =1$
2.$\displaystyle 2sin(x) cos^3(x) + 2sin^3(x)cos(x) = sin(2x)$
3.$\displaystyle (1/4)sin (4x) = sin (x) cos^3(x) - cos (x) sin^3(x)$
2.$\displaystyle 2sin(x) cos^3(x) + 2sin^3(x)cos(x)$
$\displaystyle =2sin(x)cos(x)cos^2(x)+2sin(x)sin^2(x)cos(x)$
now, group $\displaystyle 2sin(x)cos(x)$together,
and sub in $\displaystyle sin^2(x)+cos^2(x)=1$and $\displaystyle 2sin(x)cos(x)=sin(2x)$
solved!
3.$\displaystyle (1/4)sin (4x) = sin (x) cos^3(x) - cos (x) sin^3(x)$
the method is similar with question 2. try it yourself. come back to me if you still have problems.
since right-hand side of question 3 is more complicated, so we start our proof from right-hand side.
RHS: $\displaystyle sin(x)cos^3(x)-cos(x)sin^3(x)$
$\displaystyle =sin(x)cos(x)cos^2(x)-cos(x)sin(x)sin^2(x)$
$\displaystyle =sin(x)cos(x)(cos^2(x)-sin^2(x))$
$\displaystyle =(1/2)sin(2x)cos(2x)$
$\displaystyle =(1/4)sin(4x)$
so, LHS=RHS, solved!