Sin²12+Sin²21 Sin²39+Sin²48=1+Sin²9+Sin²18

LHS$={\sin }^{2}12+{\sin }^{2}21+{\sin }^{2}39+{\sin }^{2}48$

$=\frac{1-\cos 24}{2}+\frac{1-\cos 42}{2}+\frac{1-\cos 78}{2}+\frac{1-\cos 96}{2}$

$=2-\frac{\cos 24+\cos 96}{2}-\frac{\cos 42+\cos 78}{2}$

$=2-\cos \left(\frac{24+96}{2}\right)\cos \left(\frac{24-96}{2}\right)-\cos \left(\frac{42+78}{2}\right)\cos \left(\frac{42-78}{2}\right)$

$=2-\cos 60\cos 36-\cos 60\cos 18$

$=2-\frac{\cos 36+\cos 18}{2}$

$=1+\frac{2-\left(\cos 36+\cos 18\right)}{2}$

$=1+\frac{1-\cos 18}{2}+\frac{1-\cos 36}{2}$

$=1+{\sin }^{2}9+{\sin }^{2}18$

Hence Proved.