$\frac{\sin 5\theta }{\sin \theta }=$

We know that,

$\sin n\theta ={}^n{C}_1\cdot {\cos }^{n-1}\theta \cdot \sin \theta -{}^n{C}_3\cdot {\cos }^{n-3}\theta \cdot {\sin }^3\theta +\cdots$

$\Rightarrow \sin 5\theta =5{\cos }^4\theta \sin \theta -10{\cos }^2\theta \cdot {\sin }^3\theta +{\sin }^5\theta$

$\Rightarrow \frac{\sin 5\theta }{\sin \theta }=5{\cos }^4\theta -10{\cos }^2\theta {\sin }^2\theta +{\sin }^4\theta$

$=5{\cos }^4\theta -10{\cos }^2\theta (1-{\cos }^2\theta )+{(1-{\cos }^2\theta )}^2$

$=5{\cos }^4\theta -10{\cos }^2\theta +10{\cos }^4\theta +1+{\cos }^4\theta -2{\cos }^2\theta$

$=16{\cos }^4\theta -12{\cos }^2\theta +1$