A) \[\frac{{{e}^{2}}+1}{2e}\]
B) \[\frac{{{e}^{4}}+1}{2{{e}^{2}}}\]
C) \[\frac{{{({{e}^{2}}-1)}^{2}}}{2{{e}^{2}}}\]
D) \[\frac{{{({{e}^{2}}+1)}^{2}}}{2{{e}^{2}}}\]
E) \[\frac{{{({{e}^{2}}-1)}^{2}}}{4{{e}^{2}}}\]
Correct Answer: C
Solution :
Now, \[{{e}^{2}}=1+\frac{2}{1!}+\frac{{{2}^{2}}}{2!}+\frac{{{2}^{3}}}{3!}+.....\] and \[{{e}^{-2}}=1-\frac{2}{1!}+\frac{{{2}^{2}}}{2!}-\frac{{{2}^{3}}}{3!}+....\] \[\Rightarrow \] \[{{e}^{2}}+{{e}^{-2}}=2\left[ 1+\frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+..... \right]\] \[\Rightarrow \] \[\frac{{{e}^{2}}+{{e}^{-2}}}{2}-1=\left[ \frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+..... \right]\] \[\Rightarrow \] \[\frac{{{e}^{4}}+1-2{{e}^{-2}}}{2{{e}^{2}}}=\left[ \frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+..... \right]\] \[\Rightarrow \] \[\frac{{{({{e}^{2}}-1)}^{2}}}{2{{e}^{2}}}=\frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+.....\]
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