JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Integral power of iota, Algebraic operations and Equality of complex numbers

  • question_answer \[A+iB\] form of \[\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cot u+i)(1+i\tan v)}\] is [Roorkee 1980]

    A) \[\sin u\cos v\,[\cos (x+y-u-v)+i\sin (x+y-u-v)]\]

    B) \[\sin u\cos v\,[\cos (x+y+u+v)+i\sin (x+y+u+v)]\]

    C) \[\sin u\cos v\,[\cos (x+y+u+v)-i\sin (x+y+u+v)]\]

    D) None of these

    Correct Answer: A

    Solution :

    L.H.S. \[=\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cos u+i\sin u)(\cos v+i\sin v)}\]\[\sin u\cos v\] \[=\sin u\cos v[\cos (x+y-u-v)+i\sin (x+y-u-v)]\]

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