Calculate the pollowing without the use of the calculator \[ \frac{\sin ^{2} 10^{\circ}+\sin ^{2} 100^{\circ}-\cos ^{2} 200^{\circ}}{\sin \left(-20^{\circ}\right) \cdot \cos 250^{\circ}} \]

To solve the expression \[ \frac{\sin^{2}10^{\circ} + \sin^{2}100^{\circ} - \cos^{2}200^{\circ}}{\sin(-20^{\circ}) \cdot \cos 250^{\circ}}, \] we begin by simplifying each part. ### Step 1: Simplify the numerator Firstly, we know: \[ \sin 100^{\circ} = \sin (90^{\circ} + 10^{\circ}) = \cos 10^{\circ}. \] Thus, \[ \sin^{2} 100^{\circ} = \cos^{2} 10^{\circ}. \] Now, we can write the numerator: \[ \sin^{2} 10^{\circ} + \sin^{2} 100^{\circ} = \sin^{2} 10^{\circ} + \cos^{2} 10^{\circ} = 1. \] Next, we simplify \(\cos^{2} 200^{\circ}\): \[ \cos 200^{\circ} = \cos(180^{\circ} + 20^{\circ}) = -\cos 20^{\circ, \] so, \[ \cos^{2} 200^{\circ} = \cos^{2} 20^{\circ}. \] Thus, the whole numerator becomes \[ 1 - \cos^{2} 200^{\circ} = 1 - \cos^{2} 20^{\circ} = \sin^{2} 20^{\circ}. \] ### Step 2: Simplify the denominator We now evaluate the denominator: \[ \sin(-20^{\circ}) = -\sin 20^{\circ, \] and \[ \cos 250^{\circ} = \cos(270^{\circ} - 20^{\circ}) = -\sin 20^{\circ. \] Multiplying these, we find: \[ \sin(-20^{\circ}) \cdot \cos 250^{\circ} = -\sin 20^{\circ} \cdot (-\sin 20^{\circ}) = \sin^{2} 20^{\circ}. \] ### Step 3: Combine simplified numerator and denominator Now, we can express the original expression using our simplifications: \[ \frac{\sin^{2} 20^{\circ}}{\sin^{2} 20^{\circ}} = 1. \] Thus, the final result is: \[ \boxed{1}. \]