# Simplifying a trigonometric expression

1. Jul 20, 2013

### Saitama

1. The problem statement, all variables and given/known data
If $\displaystyle \tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\tan^3\left( \frac{\pi}{4}+\frac {x}{2} \right)$, prove that $$\frac{\sin y}{\sin x}=\frac{3+\sin^2 x}{1+3\sin^2x}$$

2. Relevant equations

3. The attempt at a solution
$$\tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\frac{1+\tan (y/2)}{1-\tan(y/2)}=\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}=\frac{1+\sin y}{\cos y}$$
Similarly,$$\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)=\frac{1+\sin x}{\cos x}$$
Plugging them,
$$\frac{1+\sin y}{\cos y}=\left(\frac{1+\sin x}{\cos x}\right)^3$$
Stuck here. Need a few hints to proceed further.

Any help is appreciated. Thanks!

2. Jul 20, 2013

### tiny-tim

Hi Pranav-Arora!
That's not the only way of simplifying that, is it?

(and remember, you want all sins and no coss)

3. Jul 20, 2013

### Saitama

$$\tan\left(\frac{\pi}{4}+\frac{y}{2}\right)=\frac{\sin(\pi/4+y/2)}{\sin(\pi/4-y/2)}$$
Would that help?

4. Jul 20, 2013

### tiny-tim

no, i mean find another simplification for $$\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}$$

5. Jul 20, 2013

### Saitama

I think I still cannot follow your hint. Would the following work?
$$\frac{\cos (y/2)+\sin (y/2)}{\cos (y/2)-\sin(y/2)}=\sqrt{\frac{1+\sin y}{1-\sin y}}$$

6. Jul 20, 2013

### tiny-tim

yes!!!

keep going!!!!​

7. Jul 20, 2013

### Saitama

Thanks a lot tiny-tim! I have proved it now.