g) \( \frac{\tan \theta}{\tan 2 \theta}=\frac{\cos ^{2} \theta-\sin ^{2} \theta}{1+\cos 2 \theta} \) h) \( \frac{\cos 2 \theta+\cos \theta+1}{\sin \theta+\sin 2 \theta}=\frac{1}{\tan \theta} \) i) \( \frac{\sin 2 x-\sin x}{\cos 2 x+\cos x}=\frac{\sin x}{\cos x+1} \) j) \( \cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta \)

The given equations involve trigonometric identities that can be traced back to the work of ancient Greek mathematicians. For instance, the identity \( \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \) is a specific case derived from their exploration of angles and periodic functions. These concepts were crucial not only for astronomy but also for understanding harmonic motion in their time—a blend of science and art! In practical terms, the trigonometric identities play a key role in fields such as engineering and physics. For example, the identity relating \( \tan \theta \) and \( \tan 2\theta \) helps simplify complex problems involving wave functions and oscillations. By transforming tangential relationships, engineers can analyze forces, vibrations, and other phenomena with greater ease, making these identities invaluable tools in real-world applications!