Question

It is given that $\sin {54}^o=\frac{\sqrt{5}+1}{4}$, the value of $[64{\sin }^4{27}^o-64{\sin }^2{27}^o+6{]}^6+345[{\sin }^2{27}^o+{\sin }^2{63}^o]$ is equal to

Answer 1

Given $\sin {54}^0=\frac{\sqrt{5}+1}{4}$
Consider, $[64{\sin }^4{27}^o-64{\sin }^2{27}^o+6{]}^6+345[{\sin }^2{27}^o+{\sin }^2{63}^o]$
$=\left[64{\sin }^2{27}^o\right({\sin }^2{27}^o-1)+6{]}^6+345[{\sin }^2({90}^o-{63}^o)+{\sin }^2{63}^o]$
$=[-64{\sin }^2{27}^o{\cos }^2{27}^o+6{]}^6+345[{\cos }^2{63}^o+{\sin }^2{63}^o]$
$=[-16{\sin }^2{54}^o+6{]}^6+345$
$=[-(\sqrt{5}+1{)}^2+6{]}^6+345$
$=8000+345=8345$