still testing

1 files changed, 16 insertions, 1 deletions

diff --git a/Fourier Series.page b/Fourier Series.page
index 4cf9014..1298bc9 100644
--- a/Fourier Series.page
+++ b/Fourier Series.page
@@ -8,12 +8,27 @@ To show that Fourier series is plausible, let us consider some arbitrary trignom
 $1.\quad\sin^2(x) =  ?$  
    
 Based on the double angle formula,  
-$\cos(2x) = 1 - 2 \sin^2(x)$  
+  
+$\qquad\cos(2x) = 1 - 2 \sin^2(x)$  
   
 Rearranging,  
   
 $\qquad\sin^2(x) = \frac{1-\cos(2x)}{2}$  
   
+$2.\quad\sin(2x).\cos(2x) =  ?$  
+   
+Based on the double angle formula,  
+  
+$\qquad\sin(2x) = 2\sin(x)\cos(x)$  
+  
+Rearranging,
+$$\begin{array}{ccl} 
+\sin(2x).\cos(x) & = & (\2\sin(x)\cos(x))\cos(x)\\
+ & = & 1+iy-\frac{y^{2}}{2!}-i\frac{y^{3}}{3!}+\frac{y^{4}}{4!}+i\frac{y^{5}}{5!}+\cdots\\
+ & = & (1-\frac{y^{2}}{2!}+\frac{y^{4}}{4!}+\cdots)+i(y-\frac{y^{3}}{3!}+\frac{y^{5}}{5!}-\cdots)\\
+ & = & \cos y+i\sin y\end{array}$$
+  
+$\qquad\sin^2(x) = \frac{1-\cos(2x)}{2}$ 
 ##What is the Fourier series actually?</b>
 
 ##Why is Fourier series useful? </b>