(a) Prove the following identities: (1) \( \frac{\sin 2 \theta}{\cos \theta}-\sin \theta=\tan \theta \cos \theta \) (3) \( \frac{\sin 2 \alpha}{1-\sin ^{2} \alpha}=2 \tan \alpha \) (5) \( \frac{\cos x}{\cos x-\sin x}-\frac{\cos x}{\cos x+\sin x}=\tan 2 x \) (7) \( \frac{2 \sin ^{2} \theta}{1+\cos 2 \theta^{\cos }=\tan ^{2} \theta} \) (9) \( \frac{\cos 2 x+1}{\cos x}=\frac{2 \sin x}{\tan x} \)

Let's dive into some of these trigonometric identities and break them down to construct our proofs! 1. To prove \( \frac{\sin 2 \theta}{\cos \theta} - \sin \theta = \tan \theta \cos \theta \): Starting with the left-hand side: \[ \frac{\sin 2\theta}{\cos \theta} - \sin \theta = \frac{2 \sin \theta \cos \theta}{\cos \theta} - \sin \theta = 2 \sin \theta - \sin \theta = \sin \theta. \] Now, simplifying the right-hand side: \[ \tan \theta \cos \theta = \frac{\sin \theta}{\cos \theta} \cdot \cos \theta = \sin \theta. \] Since both sides are equal, the identity is proven! 2. For \( \frac{\sin 2 \alpha}{1 - \sin^2 \alpha} = 2 \tan \alpha \): The left-hand side: \[ \frac{\sin 2\alpha}{1 - \sin^2 \alpha} = \frac{2 \sin \alpha \cos \alpha}{\cos^2 \alpha} = 2 \sin \alpha \cdot \sec^2 \alpha = 2 \tan \alpha. \] Thus, the identity holds true. 3. Next, for \( \frac{\cos x}{\cos x - \sin x} - \frac{\cos x}{\cos x + \sin x} = \tan 2x \): Let's find a common denominator: \[ \frac{\cos x (\cos x + \sin x) - \cos x (\cos x - \sin x)}{(\cos x - \sin x)(\cos x + \sin x)} = \frac{2\cos x \sin x}{\cos^2 x - \sin^2 x} = \tan 2x. \] 4. To prove \( \frac{2 \sin^2 \theta}{1 + \cos 2\theta} = \tan^2 \theta \): Using \( \cos 2\theta = 1 - 2\sin^2 \theta \), we rewrite: \[ 1 + \cos 2\theta = 2 - 2\sin^2 \theta = 2\cos^2 \theta. \] Therefore: \[ \frac{2 \sin^2 \theta}{2 \cos^2 \theta} = \tan^2 \theta. \] 5. Lastly, for \( \frac{\cos 2x + 1}{\cos x} = \frac{2 \sin x}{\tan x} \): Recall \( \cos 2x + 1 = 2\cos^2 x \): \[ \frac{2 \cos^2 x}{\cos x} = 2 \cos x. \] And for the right side: \[ \frac{2 \sin x}{\tan x} = 2 \sin x \cdot \frac{\cos x}{\sin x} = 2 \cos x. \] Both sides match! These identities hold true, proving they're solid parts of trigonometric knowledge! Keep practicing, and these will become second nature.