Prove that (1) $$ \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x-1 $$ Prove the following identities: (1) $$ \quad \frac{2+2 \cos x}{\sin 2 x}=\frac{ an x}{1-\cos x} $$ (3) $$ \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} $$

Question

Prove that (1) $$ \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x-1 $$ Prove the following identities: (1) $$ \quad \frac{2+2 \cos x}{\sin 2 x}=\frac{	an x}{1-\cos x} $$ (3) $$ \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} $$

Turnbull Martinez · Accepted Answer

The identities are proven as follows: 1. $$ \sin 4x = 4 \sin x \cos x - 8 \sin^3 x \cos x - 1 $$ is derived using double angle formulas for sine and cosine. 2. $$ \frac{2 + 2 \cos x}{\sin 2x} = \frac{	an x}{1 - \cos x} $$ is proven by simplifying both sides and using the identity $$ \sin^2 x + \cos^2 x = 1 $$. 3. $$ \sqrt{\frac{1 + \sin \alpha}{1 - \sin \alpha}} = \frac{2 \cos \alpha + \sin 2\alpha}{\cos 2\alpha + 1} $$ is proven by simplifying both sides and using trigonometric identities.

Prove that (1) \( \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x-1 \) Prove the following identities: (1) \( \quad \frac{2+2 \cos x}{\sin 2 x}=\frac{\tan x}{1-\cos x} \) (3) \( \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} \)

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Prove that (1) \( \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x-1 \) Prove the following identities: (1) \( \quad \frac{2+2 \cos x}{\sin 2 x}=\frac{\tan x}{1-\cos x} \) (3) \( \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} \)

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