Trigonometric sum.

[man111]

Calculate Sum of [tex]sin\theta.cos\theta+\frac{sin2\theta.cos^2\theta}{2!}+\frac{sin3\theta.cos^3\theta}{3!}+\frac{sin4\theta.cos^4\theta}{4!}..................................[/tex]

[kerry]

[tex]e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+.....[/tex]

Let [tex]x= e^{i \theta}cos\theta[/tex]

Then

[tex]e^{e^{i \theta}cos\theta}=1+e^{i \theta}cos\theta+\frac{e^{i2\theta}cos^2\theta}{2!}+\frac{e^{i3\theta}cos^3\theta}{3!}+\frac{e^{i4\theta}cos^4\theta}{4!}+.....=[/tex]

= [tex]\red \left( 1+cos\theta cos\theta + \frac{cos2\theta cos^2 \theta}{2!} + \frac{cos3\theta cos^3 \theta}{3!}+ \frac{cos4\theta cos^4 \theta}{4!} +..... \right)[/tex]+

+ i [tex]\blue \left( sin\theta cos\theta + \frac{sin2\theta cos^2 \theta}{2!} + \frac{sin3\theta cos^3 \theta}{3!}+ \frac{sin4\theta cos^4 \theta}{4!} +..... \right)[/tex]

But

[tex]e^{e^{i \theta}cos\theta}= e^{(cos\theta+isin\theta)cos\theta} = e^{cos^2\theta+i \frac{sin2\theta}{2}} = e^{cos^2\theta} \left( cos{ \frac{sin 2 \theta}{2}} + i sin{ \frac{sin 2 \theta }{2}}\right) =[/tex]

=[tex]\red \left( e^{cos^2\theta} cos{ \frac{sin 2 \theta}{2}}\right)[/tex] + i [tex]\blue \left(e^{cos^2\theta} sin{ \frac{sin 2 \theta }{2}}\right)[/tex]


[tex]\fbox{sin\theta cos\theta + \frac{sin2\theta cos^2 \theta}{2!} + \frac{sin3\theta cos^3 \theta}{3!}+ \frac{sin4\theta cos^4 \theta}{4!} +..... = e^{cos^2\theta} sin{ \frac{sin 2 \theta }{2}}}[/tex]

[man111]

Thanks for nice explanation.