If `2^x = cos(y/2)` and `a^x = sin y` , then `sin(y/2)` is equal to

Options

A. `1/2(a/2)^(x)`
B. `(a/2)^(x)`
C. `(a^(x))/(2)`
D. `2^(pi//2)`

Answer

To solve the problem, we start with the given equations:

1. \( 2^x = \cos\left(\frac{y}{2}\right) \)
2. \( a^x = \sin y \)

We need to find \( \sin\left(\frac{y}{2}\right) \).

### Step 1: Use the double angle formula for sine
We know the double angle formula for sine:

\[ \sin y = 2 \sin\left(\frac{y}{2}\right) \cos\left(\frac{y}{2}\right) \]

### Step 2: Substitute the known values
From the first equation, we have:

\[ \cos\left(\frac{y}{2}\right) = 2^x \]

Substituting this into the double angle formula gives:

\[ \sin y = 2 \sin\left(\frac{y}{2}\right) \cdot 2^x \]

### Step 3: Substitute \( \sin y \)
From the second equation, we know:

\[ \sin y = a^x \]

Now we can set the two expressions for \( \sin y \) equal to each other:

\[ a^x = 2 \sin\left(\frac{y}{2}\right) \cdot 2^x \]

### Step 4: Solve for \( \sin\left(\frac{y}{2}\right) \)
Rearranging the equation gives:

\[ \sin\left(\frac{y}{2}\right) = \frac{a^x}{2 \cdot 2^x} \]

This simplifies to:

\[ \sin\left(\frac{y}{2}\right) = \frac{a^x}{2^{x+1}} \]

### Final Result
Thus, we have:

\[ \sin\left(\frac{y}{2}\right) = \frac{1}{2} \cdot \frac{a^x}{2^x} \]

### Conclusion
The final answer is:

\[ \sin\left(\frac{y}{2}\right) = \frac{1}{2} \cdot \frac{a^x}{2^x} \]