Let's delve into each of these identities one by one.

33. Prove:

$\sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = \sin x$

Proof: Using the sine addition formula: $\sin(a + b) = \sin a \cos b + \cos a \sin b$ For $a = x$ and $b = \frac{\pi}{3}$: $\sin\left(x + \frac{\pi}{3}\right) = \sin x \cos\left(\frac{\pi}{3}\right) + \cos x \sin\left(\frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2}$ Similarly, $\sin\left(x - \frac{\pi}{3}\right) = \sin x \cos\left(-\frac{\pi}{3}\right) + \cos x \sin\left(-\frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}$ Adding these: $\sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = \left(\sin x \cdot \frac{1}{2} + \cos x \cdot \frac{\sqrt{3}}{2}\right) + \left(\sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2}\right) = \sin x$

34. Prove:

$\cos(x + \frac{\pi}{4}) \cos(x - \frac{\pi}{4}) = \cos^2 x - \frac{1}{2}$

Proof: Using the cosine product-to-sum identity: $\cos A \cos B = \frac{1}{2} (\cos(A+B) + \cos(A-B))$ Let $A = x + \frac{\pi}{4}$ and $B = x - \frac{\pi}{4}$: $\cos\left(x + \frac{\pi}{4}\right) \cos\left(x - \frac{\pi}{4}\right) = \frac{1}{2} \left(\cos\left(2x\right) + \cos\left(\frac{\pi}{2}\right)\right) = \frac{1}{2} \cos(2x)$ Using the identity $\cos(2x) = 2\cos^2 x - 1$: $\frac{1}{2}(2\cos^2 x - 1) = \cos^2 x - \frac{1}{2}$

35. Prove:

$\tan(x - \frac{\pi}{4}) = \frac{\tan x - 1}{\tan x + 1}$

Proof: Using the tangent subtraction formula: $\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}$ Let $a = x$ and $b = \frac{\pi}{4}$: $\tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan\left(\frac{\pi}{4}\right)}{1 + \tan x \tan\left(\frac{\pi}{4}\right)} = \frac{\tan x - 1}{1 + \tan x}$

36. Prove:

$\frac{\sin(x+y) - \sin(x-y)}{\cos(x+y) + \cos(x-y)} = \tan y$

Proof: Using sine and cosine subtraction formulas: $\sin(x+y) - \sin(x-y) = 2\cos x \sin y$ $\cos(x+y) + \cos(x-y) = 2\cos x \cos y$ Thus: $\frac{2\cos x \sin y}{2\cos x \cos y} = \frac{\sin y}{\cos y} = \tan y$

These identities are now proven. Would you like to delve deeper into any specific proof or explore related concepts? Here are some questions you might consider:

1. How can these identities be applied in solving trigonometric equations?
2. What are some graphical interpretations of these trigonometric identities?
3. Can you derive similar identities using different angles?
4. How do these identities simplify calculus problems involving trigonometric functions?
5. What are the common pitfalls when proving trigonometric identities?

Tip: Always simplify both sides of the equation independently before concluding they are equal.