Question

Evaluate ${lim}_{x\to 0}\frac{sin(1+x)-sin(1-x)}{x}$

Answer 1

${lim}_{x\to 0}\frac{sin(1+x)-sin(1-x)}{x}$

Using $sinC-sinD=2sin\left(\frac{C-D}{2}\right)cos\left(\frac{C+D}{2}\right)$

${lim}_{x\to 0}\frac{2sin\left(\frac{(1+x)-(1-x)}{2}\right)cos\left(\frac{(1+x)+(1-x)}{2}\right)}{x}$

${lim}_{x\to 0}2cos1\frac{sinx}{x}=2cos1$