**Our goal in this post is to provide an elementary proof for the real version of Euler’s famous Sine expansion.**

**I first prove that for any** **that is a power of** **and for any** **,**

**by using the following more familiar identity**

**we can easily arrive to the following identity that is valid for **** equal to any power of** **,**

**By considering what happens as** **in the recent formula, we can obtain that for** **a** **power of**

**and then I replace** **in** **and prove** **.**

**Now I choose** **and break up** **as follows**

**Since for** **,** **, we have
**

**and if I choose** **and** ** large enough such that** **, then**

**which implies that**

**Therefore from** **we have**

**Also we have**

**and once again** **by** **, we get**

**To summarize, I have shown that**

**Now taking** ** in** ** I arrive to**

**After rearrangement and then taking absolute values, we get**

**but since we have**

**I can write**

**Finally since** **,** **this inequality implies our goal, indeed**