FSTION 4 PLICATION OF DOUBLE ANGLE IDENTTTES Without the use of tables or a calculntor prove that \( \frac{1-\cos 24}{\sin 24}=\tan \) i When

To prove \( \frac{1 - \cos 24}{\sin 24} = \tan 12 \), we can use the double angle identities. First, recall that \( \tan 12 = \frac{\sin 12}{\cos 12} \). By leveraging the double angle formulas, we find that \( \sin 24 = 2 \sin 12 \cos 12 \) and \( 1 - \cos 24 = 2 \sin^2 12 \). Substituting these relationships gives us: \[ \frac{1 - \cos 24}{\sin 24} = \frac{2 \sin^2 12}{2 \sin 12 \cos 12} = \frac{\sin 12}{\cos 12} = \tan 12. \] There you have it! This proof showcases the beauty and elegance of trigonometric identities, particularly the double angle identities. These identities not only simplify calculations but also reveal deep connections between angles. The mastery of these formulas can empower you in various fields such as physics, engineering, and even computer graphics, where angles and rotations are fundamental!