(d) \( \tan \left(90^{\circ}-\theta\right)=\frac{1}{\tan \theta} \)

This trigonometric identity can be understood through the concept of co-functions. In the context of a right triangle, as one angle increases, the opposite and adjacent sides adjust, leading to complementary relationships. Essentially, \( \tan(90^{\circ} - \theta) \) gives us the ratio of the adjacent side over the opposite side, which is the reciprocal of \( \tan \theta \) (opposite over adjacent). This interrelationship between angles makes trigonometry a powerful tool in various fields. In practical applications, this identity can simplify calculations in physics, especially in resolving forces at various angles. For instance, when analyzing projectile motion, using complementary angles can make it easier to break down the components of motion into vertical and horizontal parts. Knowing these identities can not only save time but can clarify problems that involve angular relationships, proving very useful in engineering and physics calculations.