If `g(x)=sinx,x""inRandf(x)=(1)/(sinx),x"in(0,(pi)/(2))` what is (gof)(x) equal to ?

Options

A. 1
B. `(1)/(sin(sinx))`
C. `(1)/(sin^(2)(x))`
D. `sin((1)/(sinx))`

Explanation

To solve the problem, we need to find \( (g \circ f)(x) \), which means we need to evaluate \( g(f(x)) \).

### Step 1: Identify the functions
We have:
- \( g(x) = \sin x \) for \( x \in \mathbb{R} \)
- \( f(x) = \frac{1}{\sin x} \) for \( x \in \left(0, \frac{\pi}{2}\right) \)

### Step 2: Substitute \( f(x) \) into \( g(x) \)
Now, we need to find \( g(f(x)) \):
\[ g(f(x)) = g\left(\frac{1}{\sin x}\right) \]

### Step 3: Apply the function \( g \)
Since \( g(x) = \sin x \), we can substitute \( \frac{1}{\sin x} \) into \( g \):
\[ g\left(\frac{1}{\sin x}\right) = \sin\left(\frac{1}{\sin x}\right) \]

### Step 4: Write the final answer
Thus, we have:
\[ (g \circ f)(x) = \sin\left(\frac{1}{\sin x}\right) \]

### Final Answer
\[ (g \circ f)(x) = \sin\left(\frac{1}{\sin x}\right) \]

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