Evaluates :

$open vertical bar table row cell straight a plus ib space space space space space straight c plus id end cell row cell negative straight c plus id space space space space space straight a minus ib space space space end cell end table close vertical bar$

Solution

Consider $open vertical bar table row cell straight a plus ib space space space space space straight c plus id end cell row cell negative straight c plus id space space space space space straight a minus ib space space space end cell end table close vertical bar$ = (a + i b) (a - i b)-(c + i b)(- c + i d)
= (a² - i²b²) - (i² d² - c²)
= (a² +b²) - (- d² - c²) [ $because$ i² = - 1 ]
= (a² +b² +c² + d²)

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Inverse Trigonometric Functions

Evaluates :

$open vertical bar table row cell straight a plus ib space space space space space straight c plus id end cell row cell negative straight c plus id space space space space space straight a minus ib space space space end cell end table close vertical bar$

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