Question (c)! how do i know that t^5-10t^3+5t=0? !

(a) By DeMoivre's theorem, we have



On the LHS, expanding yields



Matching up real and imaginary parts, we have for (i) and (ii),




(b) By the definition of the tangent function,







(c) Setting , we have and . So



At the given value of , the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.



Remember, this is saying that



If we replace with a variable , then the above means is a root to the quadratic equation,



Also, if , then and . So by a similar argument as above, we deduce that is also a root to the quadratic equation above.

(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write



Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy



But for all , as is the case for and , so we choose the positive root.