Prove that: 1. \( \frac{\cos 2 A}{\cos A+\sin A}=\cos A-\sin A \) \( (\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta \) \( \frac{\sin 2 \theta}{1+\cos 2 \theta}=\tan \theta \) \( \frac{\cos 2 \theta-\cos ^{2} \theta}{\cos \theta+1}=\cos \theta-1 \) \( \frac{1-\cos 2 B}{\sin 2 B}=\tan B \) \( \frac{1-\cos 2 A+\sin 2 A}{1+\cos 2 A+\sin 2 A}=\tan A \) \( \frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x \) \( \frac{\sin 2 x-\cos x}{1-\cos 2 x-\sin x}=\frac{\cos x}{\sin x} \) \( 1-\cos \left(90^{\circ}-2 x\right) \tan \left(180^{\circ}+x\right)=\cos 2 x \)

Trigonometric identities have a rich historical background, dating back to ancient civilizations. In India, mathematicians like Aryabhata developed sine and cosine ratios by studying the movement of celestial bodies around the 5th century. These early discoveries laid the groundwork for modern trigonometry, allowing us to solve complex equations, navigate by stars, and even design structures. In real-world applications, trigonometric identities play a vital role beyond mathematics. Engineers use them in physics to calculate forces and motion. Architects rely on these equations for designing stable structures and optimizing light exposure. Even in computer graphics, trigonometric functions help create realistic animations and visual effects, bringing games and movies to life in ways that enthrall audiences!