TELL ME THE ANSWER FOR 15th QUESTION . PLEASE TELL  ME FAST
 
Q15. Prove that ( cos $\theta + i \sin \theta$)ⁿ = cos $\left(\mathrm{n} \mathrm{\theta }\right) +i \sin ( \mathrm{n} \mathrm{\theta })$, for all n $\in$ N by using PMI.

Dear Student,
 de Moivre's theorem by mathematical induction for natural numbers, and extending it to all integers. For an integer n, call the following statement S(n):

For n > 0, by mathematical induction S(1) is clearly true. For our hypothesis, we assume S(k) is true for some natural k. We assume now

${displaystyle left(cos x+isin xright)^{k}=cos kx+isin kx.}$

, considering S(k + 1):

${displaystyle {begin{alignedat}{2}left(cos x+isin xright)^{k+1}&=left(cos x+isin xright)^{k}left(cos x+isin xright)&=left(cos kx+isin kxright)left(cos x+isin xright)&&qquad {text{by the induction hypothesis}}&=cos kxcos x-sin kxsin x+ileft(cos kxsin x+sin kxcos xright)&=cos ,(k+1)x+isin ,(k+1)x&&qquad {text{by the trigonometric identities}}end{alignedat}}}$ Now, S(k) implies S(k + 1). Through the principle of mathematical induction, it follows that the result is true for all natural numbers.

Hope this information will clear your doubts about topic.

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.
 

Regards