37.2. \( \frac{\sin (2 x)}{1+\cos (2 x)}=\tan x \)

(0.50) - Qual é a medida do lado oposto ngulo de \( 30^{\circ} \), em um triângulo, sabendo que utros dois lados medem 2 e \( \sqrt{3} \) ?

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) \( \quad \sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{2 \cos \alpha+\sin 2 x}{\cos 2 \alpha+1} \)

Solve the following equation for \( \beta \) over the interval \( [0,2 \pi) \), giving exact answers in radian units. If an equation has no solution, enter DNE. Multiple solutions should be entered as a comma-separated list. \( \tan ^{2}(\beta)-1=-2 \tan (\beta) \) \( \beta=\square \)

Prove that \( \begin{array}{ll}\text { (1) } \sin 4 x=4 \sin x \cos x-8 \sin ^{3} x \cos x & \text { (2) } \cos 4 x=8 \cos ^{4} x-8 \cos ^{2} x+1\end{array} \)

Solve \( 5 \cos (2 \alpha)=5 \cos ^{2}(\alpha)-4 \) for all solutions \( 0 \leq \alpha<2 \pi \) \( \alpha=\square \) Give your answers accurate to at least 2 decimal places and in a list separated by commas.