If `tan(pi/4+y/2)=tan^3(pi/4+x/2)dot` Prove that `(siny)/(sinx)=(3+sin^2x)/(1+3sin^2x)` .

Explanation

To prove that \(\frac{\sin y}{\sin x} = \frac{3 + \sin^2 x}{1 + 3 \sin^2 x}\) given that \(\tan\left(\frac{\pi}{4} + \frac{y}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{x}{2}\right)\), we will follow these steps:

### Step 1: Rewrite the given equation using the tangent addition formula
Using the formula for \(\tan(a + b)\):
\[ \tan\left(\frac{\pi}{4} + \frac{y}{2}\right) = \frac{\tan\left(\frac{\pi}{4}\right) + \tan\left(\frac{y}{2}\right)}{1 - \tan\left(\frac{\pi}{4}\right) \tan\left(\frac{y}{2}\right)} \]
Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we have:
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