There’s an error in this alleged identity, because the cos(x) terms would cancel leaving 2sin²(x)=1, which is not true for all x, disproving the identity, so 2sin²(x)-1=0=(sin(x)√2-1)(sin(x)√2+1). Therefore sin(x)=±1/√2=±√2/2, making x=±45° or ±π/4. Other values of x can be found by adding 360n° or 2πn to these values, where n is any integer. So this can only be solved as an equation and not proved as an identity. However, the solution demonstrates factoring or taking the square root.