If `(sinx)/(1+cosx)+(sinx)/(1-cosx)=4,and0^(@)ltxlt90^(@)` , then find the value of x .

Options

A. `10^(@)`
B. `15^(@)`
C. `45^(@)`
D. `30^(@)`

Answer

To solve the equation

\[ \frac{\sin x}{1 + \cos x} + \frac{\sin x}{1 - \cos x} = 4 \]

we can follow these steps:

### Step 1: Combine the fractions
First, we can take \(\sin x\) as a common factor:

\[ \sin x \left( \frac{1}{1 + \cos x} + \frac{1}{1 - \cos x} \right) = 4 \]

### Step 2: Find a common denominator
Next, we need to combine the two fractions inside the parentheses. The common denominator is \((1 + \cos x)(1 - \cos x)\):

\[ \frac{1 - \cos x + 1 + \cos x}{(1 + \cos x)(1 - \cos x)} = \frac{2}{(1 + \cos x)(1 - \cos x)} \]

### Step 3: Substitute back into the equation
Now we substitute this back into the equation:

\[ \sin x \cdot \frac{2}{(1 + \cos x)(1 - \cos x)} = 4 \]

### Step 4: Simplify the equation
We can simplify the equation further:

\[ \frac{2 \sin x}{(1 + \cos x)(1 - \cos x)} = 4 \]

### Step 5: Cross-multiply
Cross-multiplying gives us:

\[ 2 \sin x = 4 (1 + \cos x)(1 - \cos x) \]

### Step 6: Simplify the right-hand side
Using the identity \(1 - \cos^2 x = \sin^2 x\):

\[ 2 \sin x = 4 \sin^2 x \]

### Step 7: Rearrange the equation
Rearranging gives us:

\[ 4 \sin^2 x - 2 \sin x = 0 \]

### Step 8: Factor the equation
Factoring out \(2 \sin x\):

\[ 2 \sin x (2 \sin x - 1) = 0 \]

### Step 9: Solve for \(\sin x\)
Setting each factor to zero gives us:

1. \(2 \sin x = 0 \Rightarrow \sin x = 0\) (not valid in the interval \(0 < x < 90\))
2. \(2 \sin x - 1 = 0 \Rightarrow \sin x = \frac{1}{2}\)

### Step 10: Find the angle \(x\)
The value of \(x\) for which \(\sin x = \frac{1}{2}\) in the interval \(0 < x < 90\) degrees is:

\[ x = 30^\circ \]

Thus, the value of \(x\) is \(30^\circ\).

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