# TELL ME THE ANSWER FOR 15th QUESTION . PLEASE TELL  ME FAST Q15. Prove that ( cos )n = cos , 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.