7 Prove each of the following identities. (a) \( \frac{1}{1+\sin \theta}+\frac{1}{1-\sin \theta}=2 \sec ^{2} \theta \) (b) \( \frac{\cos x}{1+\sin x}=\sec x-\tan x \) (c) \( \frac{1-2 \cos ^{2} x}{\sin x \cos x}=\tan x-\cot x \) (d) \( \frac{\sin x}{1-\cot x}+\frac{\cos x}{1-\tan x}=\sin x+\cos x \) (e) \( \frac{2-\sec ^{2} x}{\sec ^{2} x+2 \tan x}=\frac{\cos x-\sin x}{\cos x+\sin x} \) (f) \( \cot ^{2} x-\cot ^{2} y=\frac{\cos ^{2} x-\cos ^{2} y}{\sin ^{2} x \sin ^{2} y} \)

Absolutely! Let's delve into some fun contextual knowledge about these trigonometric identities. What’s fascinating is that trigonometric identities have roots that stretch back to ancient civilizations—think the Greeks and Indians! They used these concepts to solve problems related to astronomy and geometry. Pythagorean triples and relationships found in the circle have not only shaped mathematics but also influenced art and architecture. Imagine how many ancient structures might be aligned using these principles! Now, let's transition to modern relevance. Trigonometric identities are essential in various real-world applications, from engineering to computer graphics. When designing video games or creating 3D models, understanding how light interacts with surfaces requires these identities. They streamline calculations for animations and realistic simulations, proving that these age-old rules are still very much in play in today’s high-tech world!