A) \[\frac{\sqrt{5}+1}{8}\]
B) \[\frac{\sqrt{5}-1}{8}\]
C) \[\frac{\sqrt{5}+1}{16}\]
D) None of these
Correct Answer: A
Solution :
\[{{\cos }^{2}}{{48}^{o}}-{{\sin }^{2}}{{12}^{o}}=\frac{1}{2}(2{{\cos }^{2}}{{48}^{o}}-2{{\sin }^{2}}{{12}^{o}})\] \[=\frac{1}{2}[(1+\cos {{96}^{o}})-(1-\cos {{24}^{o}})]\] \[=\frac{1}{2}[(\cos {{96}^{o}}+\cos {{24}^{o}})]\] \[=\frac{1}{2}\left[ 2\cos \left( \frac{{{96}^{o}}+{{24}^{o}}}{2} \right)\cos \left( \frac{{{96}^{o}}-{{24}^{o}}}{2} \right) \right]\] \[=\cos {{60}^{o}}\cos {{36}^{o}}\] \[=\frac{1}{2}\times \left( \frac{\sqrt{5}+1}{4} \right)\] \[=\frac{\sqrt{5}+1}{8}\]
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