Uprostiti sledeći izraz:

${\left(\frac{{2}^{x}+{2}^{-x}}{2}\right)}^{2}-{\left(\frac{{2}^{x}-{2}^{-x}}{2}\right)}^{2}+1$

${\left(\frac{{2}^{x}+{{2}}^{-x}}{2}\right)}^{2}-{\left(\frac{{2}^{x}-{{2}}^{-x}}{2}\right)}^{2}+1$

$=\frac{{\left({2}^{x}+\frac{1}{{2}^{x}}\right)}^{2}}{{2}^{2}}-\frac{{\left({2}^{x}-\frac{1}{{2}^{x}}\right)}^{2}}{{2}^{2}}+1$

$=\frac{{\left(\frac{{{2}}^{{x}}{·}{{2}}^{{x}}+1}{{2}^{x}}\right)}^{2}}{\frac{{2}^{2}}{1}}-\frac{{\left(\frac{{{2}}^{{x}}{·}{{2}}^{{x}}-1}{{2}^{x}}\right)}^{2}}{\frac{{2}^{2}}{1}}+1$

$=\frac{\frac{{\left({{2}}^{2x}+1\right)}^{2}}{{\left({2}^{x}\right)}^{2}}}{\frac{{2}^{2}}{1}}-\frac{\frac{{\left({{2}}^{2x}-1\right)}^{2}}{{\left({2}^{x}\right)}^{2}}}{\frac{{2}^{2}}{1}}+1$

$=\frac{{\left({2}^{2x}+1\right)}^{2}}{{{2}}^{{2}}{·}{{2}}^{2x}}-\frac{{\left({2}^{2x}-1\right)}^{2}}{{{2}}^{{2}}{·}{{2}}^{2x}}+1$

$=\frac{{\left({2}^{2x}+1\right)}^{2}}{{{2}}^{2x+2}}-\frac{{\left({2}^{2x}-1\right)}^{2}}{{{2}}^{2x+2}}+1$

$=\frac{{\left({2}^{2x}+1\right)}^{2}-{\left({2}^{2x}-1\right)}^{2}}{{2}^{2x+2}}+1$

$=\frac{{\left({2}^{2x}\right)}^{{2}}+{2}{·}{{2}}^{2x}·1+1-\left[{\left({2}^{2x}\right)}^{{2}}-{{2}}^{2x}{·}{2}+1\right]}{{2}^{·+2}}+1$

$=\frac{{{2}}^{4x}+{2}{·}{{2}}^{2x}·1+1-\left[{{2}}^{4x}-{{2}}^{2x+1}+1\right]}{{2}^{·+2}}+1$

$=\frac{{{2}}^{4x}{+}{{2}}^{2x+1}{+}{1}{-}{{2}}^{4x}{+}{{2}}^{2x+1}{-}{1}}{{2}^{2x-2}}+1$

$=\frac{\overline{){2}^{4x}}{+}{{2}}^{2x+1}\overline{)+1}\overline{)-{2}^{4x}}{+}{{2}}^{2x+1}\overline{)-}{1}}{{2}^{2x-2}}+1$

$=\frac{{2}{·}{{2}}^{2x+1}}{{2}^{2x+2}}+1$

$=\frac{{{2}}^{2x+1+1}}{{2}^{2x+2}}+1$

$=\frac{{2}^{2x+2}}{{2}^{2x+2}}+1$

$=1+1$

$=2$