 Hi and welcome to the session. I am Asha and I am going to help you with the following question that says for any two complex numbers z1 and z2 proves that real part of z1 into z2 is equal to real part of z1 into real part of z2 minus imaginary part of z1 into imaginary part of z2. Now as we know a complex number is of tip form 8 plus iota b where a is the real part of z b is called the imaginary part of z. So this idea we are going to use to prove the above problem. So this will be our t idea. Let us now begin with the solution and let z1 is equal to a plus iota b z2 be equal to c plus iota d. Now z1 into z2 equal to a plus iota b into c plus iota d which is further equal to ac plus iota times ad plus iota times bc plus iota square bt which is further equal to the iota square bt can be written as minus 1 into bt since iota square is equal to minus 1 and minus into plus is minus. So this equation can further be written as ac minus bt plus now taking i common from the these two terms can be written as ad plus bc. So the real part of z1 into z2 will be ac minus bt. Let this be equation number one. Now let us find the real part of z1 it is a real part of z2 is c imaginary part of z1 is b and imaginary part of z2 is t. Now let us find the value of real part of z1 into real part of z2 minus imaginary part of z1 into imaginary part of z2. The real part of z1 is a real part of z2 is c minus imaginary part of z1 is b and imaginary part of z2 is d which is equal to real part of z1 into z2 this is from equation one this implies that real part of z1 into z2 is equal to real part of z1 into real part of z2 minus imaginary part of z1 into imaginary part of z2. So this completes the solution hope you enjoyed it take care and have a good day.