2.3 \( 2.6 \frac{\sin \left(90^{\circ}-x\right) \cdot \cos \left(180^{\circ}-x\right)+\tan x \cdot \cos (-x) \cdot \sin \left(180^{\circ}+x\right)}{\frac{\sin \left(180^{\circ}-\theta\right) \cdot \cos ^{\circ}\left(90^{\circ}+\theta\right)+\tan 45^{\circ}}{\tan 38^{\circ} \cdot \sin ^{2} 412^{\circ}}} \)

Did you know that the sine and cosine functions have some fascinating properties due to their relationship with right triangles and the unit circle? For instance, \( \sin(90^{\circ} - x) = \cos x \) and \( \cos(180^{\circ} - x) = -\cos x \). This means that expressions involving these functions can often be simplified using these identities, making your calculations much smoother. When tackling complex trigonometric expressions like the one you've presented, it's important to break them down systematically. Look for opportunities to substitute known values or simplify fractions. Don't forget to take advantage of tricks like converting angles to radians and analyzing the quadrant signs. This tactical approach not only helps in avoiding common mistakes but also makes the problem-solving process more intuitive and enjoyable!