The value of $$sin^{2} 65^{\circ} + sin^{2} 25^{\circ} + cos^{2} 35^{\circ} + cos^{2} 55^{\circ}$$ is

$$sin^{2} 65^{\circ} + sin^{2} 25^{\circ} + cos^{2} 35^{\circ} + cos^{2} 55^{\circ}$$

= $$sin^{2} (90-25) + sin^{2} 25 + cos^{2} (90-55) + cos^{2} 55$$

we know $$sin (90 - \theta)$$ = $$cos \theta$$

& , $$cos (90 - \theta)$$ = $$sin \theta$$

using above identities ,

$$sin^{2} (90-25) + sin^{2} 25 + cos^{2} (90-55) + cos^{2} 55$$ = $$cos^{2} (25) + sin^{2} (25) + sin^{2} (55) + cos^{2} (55)$$

using $$sin^2 \theta$$ + $$cos^2 \theta$$ = 1

$$cos^{2} (25) + sin^{2} 25 + sin^{2} (55) + cos^{2} 55$$ = 1 + 1 = 2