Question

If z = r( cos theta + i sin theta ) , then value of z/z + z/z is

Answer 1

Given :  z = r( cosθ + i sin θ )

To Find :  $\dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}$

Solution:

z = r( cosθ + i sin θ )

$\overline{z}$ = r( cosθ -  i sin θ )

$\dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}$  

=  r( cosθ + i sin θ ) /r ( cosθ -  i sin θ )    + r( cosθ -  i sin θ ) /r( cosθ +  i sin θ )

=  ( cosθ + i sin θ ) / ( cosθ -  i sin θ )    +  ( cosθ -  i sin θ ) / ( cosθ +  i sin θ )

=  (  ( cosθ + i sin θ )² + ( cosθ - i sin θ )²) /(cos²θ -  i² sin² θ)

=  (  2( cos²θ + i² sin² θ )) /(cos²θ -  i² sin² θ)

i²  = - 1

=  (  2( cos²θ -  sin² θ )) /(cos²θ +  sin² θ)

cos²θ +  sin² θ = 1

cos²θ -  sin² θ = cos2θ

= 2 cos2θ

$\dfrac{z}{\overline{z}} +\dfrac{\overline{z}}{z}$ = 2 cos2θ

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