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Question Number 60056 by bhanukumarb2@gmail.com last updated on 17/May/19

Commented by maxmathsup by imad last updated on 17/May/19

$$weseethatS=\sum _{k=0}^{\mathrm{\infty }}\frac{sin\left(k\theta \right){cos}^k\theta }{k!}\Rightarrow S=Im\left(\sum _{k=0}^{\mathrm{\infty }}\frac{e^{ik\theta }{cos}^k\theta }{k!}\right)but$$$$\sum _{k=0}^{\mathrm{\infty }}\frac{e^{ik\theta }{cos}^k\theta }{k!}=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(e^{i\theta }cos\theta \right)}^k}{k!}=e^{(e^{i\theta cos\theta )}}=e^{cos\left(\theta cos\theta \right)+isin\left(\theta cos\theta \right))}$$$$=e^{cos\left(\theta cos\theta \right)}\{cos(sin\left(\theta cos\theta \right)+isin\left(sin\left(\theta cos\theta \right)\right)\}\Rightarrow$$$$S=e^{cos\left(\theta cos\theta \right)}sin\left(sin\left(\theta cos\theta \right)\right).$$

Answered by tanmay last updated on 17/May/19

$$p=cos\theta cos\theta +\frac{cos2\theta {cos}^2\theta }{2!}+\frac{cos3\theta {cos}^3\theta }{3!}+...$$$$q=sin\theta cos\theta +\frac{sin2\theta {cos}^2\theta }{2!}+\frac{sin3\theta {cos}^3\theta }{3!}+..$$$$p+iq$$$$=e^{i\theta }cos\theta +e^{i2\theta }\times \frac{{cos}^2\theta }{2!}+\frac{e^{i3\theta }\times {cos}^3\theta }{3!}+...$$$$t=e^{i\theta }cos\theta$$$$p+iq=t+\frac{t^2}{2!}+\frac{t^3}{3!}+...$$$$=e^t-1$$$$=e^{e^{i\theta }cos\theta }-1$$$$=e^{cos\theta (cos\theta +isin\theta )}-1$$$$=e^{{cos}^2\theta +isin\theta cos\theta }-1$$$$=e^{{cos}^2\theta }\times \left(e^{isin\theta cos\theta }\right)-1$$$$=e^{{cos}^2\theta }\times e^{\frac{isin2\theta }{2}}-1$$$$=e^{{cos}^2\theta }[cos\left(\frac{sin2\theta }{2}\right)+isin\left(\frac{sin2\theta }{2}\right)]-1$$$$=e^{{cos}^2\theta }\left[cos\left(\frac{sin2\theta }{2}\right)\right]-1+{ie}^{{cos}^2\theta }sin\left(\frac{sin2\theta }{2}\right)$$$$\mathit{S}op=e^{{cos}^2\theta }\left[cos\left(\frac{sin2\theta }{2}\right)\right]-1\leftarrow realpart$$$$q=e^{{cos}^2\theta }sin\left(\frac{sin2\theta }{2}\right)\leftarrow imaginarypart$$$$\mathit{S}o\mathit{r}\mathit{e}\mathit{q}\mathit{u}\mathit{i}\mathit{r}\mathit{e}\mathit{d}\mathit{a}\mathit{n}\mathit{s}\mathit{w}\mathit{e}\mathit{r}\mathit{i}\mathit{s}$$$$\mathit{q}={\mathit{e}}^{{\mathit{c}\mathit{o}\mathit{s}}^2\theta }sin\left(\frac{sin2\theta }{2}\right)$$$$plscheck....$$$$$$

Commented by bhanukumarb2@gmail.com last updated on 17/May/19

$$rightsirthanku$$

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