If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?

If (cosθ + i sinθ)² = x + iy, then show that x²+y² =1

If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0

If 1, ω, ω² are the cube roots of unity show that (1+ω²)³ - (1+ω)³ = 0

Find the modules of (1+ 3i)³