\displaystyle \text{Prove}\ \text{that}\ \ \ \frac{1}{2}\sqrt{{4{{{\sin }}^{2}}{{{36}}^{\circ }}-1}}=\cos {{72}^{\circ }}.
\displaystyle \text{Solution}
\displaystyle \ \ \ \cos {{72}^{\circ }}=2{{\cos }^{2}}{{36}^{\circ }}-1\ \ \ ........(1)
\displaystyle \ \ \ \cos {{144}^{\circ }}=2{{\cos }^{2}}{{72}^{\circ }}-1
\displaystyle \ \ \ -\cos {{36}^{\circ }}=2{{\cos }^{2}}{{72}^{\circ }}-1
\displaystyle \ \ \ \ \ \cos {{36}^{\circ }}=1-2{{\cos }^{2}}{{72}^{\circ }}\ \ .......(2)
\displaystyle (1)+(2)\Rightarrow \ \ \ \cos {{72}^{\circ }}+\cos {{36}^{\circ }}=2\ (\ {{\cos }^{2}}{{36}^{\circ }}-{{\cos }^{2}}{{72}^{\circ }})
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cos {{36}^{\circ }}-\cos {{72}^{\circ }}=\frac{1}{2}
\displaystyle (2)-(1)\Rightarrow \ \ \ \cos {{36}^{\circ }}-\cos {{72}^{\circ }}=2-2\ (\ {{\cos }^{2}}{{36}^{\circ }}+{{\cos }^{2}}{{72}^{\circ }})
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2}=2-2\ (\ {{\cos }^{2}}{{36}^{\circ }}+{{\cos }^{2}}{{72}^{\circ }})
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ (\ {{\cos }^{2}}{{36}^{\circ }}+{{\cos }^{2}}{{72}^{\circ }})=\frac{3}{2}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\cos }^{2}}{{36}^{\circ }}+{{\cos }^{2}}{{72}^{\circ }}=\frac{3}{4}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1-{{\sin }^{2}}{{36}^{\circ }}+{{\cos }^{2}}{{72}^{\circ }}=\frac{3}{4}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\sin }^{2}}{{36}^{\circ }}-{{\cos }^{2}}{{72}^{\circ }}=\frac{1}{4}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4{{\sin }^{2}}{{36}^{\circ }}-4{{\cos }^{2}}{{72}^{\circ }}=1
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4{{\sin }^{2}}{{36}^{\circ }}-1=4{{\cos }^{2}}{{72}^{\circ }}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sqrt{{4{{{\sin }}^{2}}{{{36}}^{\circ }}-1}}=2\cos {{72}^{\circ }}
\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore \ \ \frac{1}{2}\sqrt{{4{{{\sin }}^{2}}{{{36}}^{\circ }}-1}}=\cos {{72}^{\circ }}
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