If si$n^4$θ − co$s^4$θ = 1/3, then si$n^2$θ equals:

si$n^4$θ − co$s^4$θ factorises as (si$n^2$θ − co$s^2$θ)(si$n^2$θ + co$s^2$θ) = si$n^2$θ − co$s^2$θ. So si$n^2$θ − co$s^2$θ = 1/3 while si$n^2$θ + co$s^2$θ = 1; adding gives 2si$n^2$θ = 4/3, so si$n^2$θ = 2/3. Reading the first equation as si$n^2$θ = 1/3 skips the second one.