If 1+i=(x+i y)(u+i v) , then ⁻¹ ( y x )+ ⁻¹ ( u v ) has the value

Step-by-step solution

Given, equation is gathered 1+i=(x+i y)(u+i v) 1+i=(x u-y v)+i(x v+y u) gathered Comparing real and imaginary parts, we get aligned &x u-y v=1 ...(i) &x v+y u=1 ...(ii) aligned Multiply Eq. (i) by u and Eq. (ii) by v and then adding, we get x (u²+v² )=u+v x= u+v u²+v² From Eq. (i) y= x u-1 v = u-v u²+v² (Substituting the value of x ) Now, ⁻¹(y / x)+ ⁻¹(u / v) aligned ⁻¹ ( u-v u+v )+ ⁻¹(u / v) &= ⁻¹ ( 1- v u 1+ v u )+ ⁻¹(u / v) &= ⁻¹(1)- ⁻¹ ( v u )+ ⁻¹ ( v u ) &=n + / 4, n I aligned