Question

he complex formsof the sine and cosine functions are:ieexixix2sinand2cosixixeex, where eis the Euler e, and 1i.Differentiate the complex form of cosxto showthatxxdxdsin)(cos.[Hints:The Quotient Rule may be used, but it is not necessaryif you factor out a constant first.√−1=iis a constant. In fact, you can use it to multiply a fraction by iiif that helps...]

Answer 1

Answer:

$\cos'(z) = -\sin(z)$

Step-by-step explanation:

According to the information given by the problem

$\sin(z) = {\displaystyle \frac{e^{iz} - e^{-iz} }{2i} }$

$\cos(z) = {\displaystyle \frac{e^{iz} + e^{-iz} }{2} }$

Now, if you compute the derivative of $\cos$ you get that

$\cos'(z) = {\displaystyle \frac{ ie^{iz}-i e^{iz} }{2} } = {\displaystyle \frac{ i ( e^{iz}- e^{iz} )}{2} }\\\\= {\displaystyle \frac{ i ( e^{iz}- e^{iz} )}{2} } *\frac{i}{i} }\\\\= {\displaystyle - \frac{ e^{iz}- e^{iz} }{2i} } = -\sin(z)$