Evaluate : 4-5 4+5 + 4+5 4-5 - Vidyadip

Question

Evaluate : $\frac{4-\sqrt{5} }{4+\sqrt{5} }+\frac{4+\sqrt{5} }{4-\sqrt{5} }$

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Answer

$ \frac{4-\sqrt{5}}{4+\sqrt{5 }}+\frac{4+\sqrt{5}}{4-\sqrt{5}}$
$ =\frac{4-\sqrt{5}}{4+\sqrt{5}} \times \frac{4-\sqrt{5}}{4-\sqrt{5}}+\frac{4+\sqrt{5}}{4-\sqrt{5}} \times \frac{4+\sqrt{5}}{4+\sqrt{5}}$
$ =\frac{(4-\sqrt{5})^2}{(4)^2-(\sqrt{5})^2}+\frac{(4+\sqrt{5})^2}{(4)^2-(\sqrt{5})^2}$
$ =\frac{16+5-8 \sqrt{5}}{16-5}+\frac{16+5+8 \sqrt{5}}{16-5}$
$ =\frac{21-8 \sqrt{5}}{11}+\frac{21+8 \sqrt{5}}{11}$
$ =\frac{21-8 \sqrt{5}+21+8 \sqrt{5 }}{11}$
$ =\frac{42}{11}=3 \frac{9}{11}$

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