If x is acute , then `sqrt((1+sinx)/(1-sinx))` is

Options

A. sec x + tan x
B. cosec x + cot x
C. sec x + cosec x
D. tan x + cot x

Answer

To solve the expression \( \sqrt{\frac{1 + \sin x}{1 - \sin x}} \) when \( x \) is acute, we can follow these steps:

### Step 1: Write the expression
We start with the expression:
\[ y = \sqrt{\frac{1 + \sin x}{1 - \sin x}} \]

### Step 2: Rationalize the expression
To simplify this expression, we can multiply the numerator and the denominator by \( 1 + \sin x \):
\[ y = \sqrt{\frac{(1 + \sin x)(1 + \sin x)}{(1 - \sin x)(1 + \sin x)}} \]

### Step 3: Simplify the numerator and denominator
The numerator becomes:
\[ (1 + \sin x)^2 = 1 + 2\sin x + \sin^2 x \]
The denominator simplifies using the difference of squares:
\[ (1 - \sin x)(1 + \sin x) = 1^2 - (\sin x)^2 = 1 - \sin^2 x \]
Using the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the denominator:
\[ 1 - \sin^2 x = \cos^2 x \]

### Step 4: Substitute back into the expression
Now we can substitute back into our expression:
\[ y = \sqrt{\frac{1 + 2\sin x + \sin^2 x}{\cos^2 x}} \]

### Step 5: Separate the square root
We can separate the square root:
\[ y = \frac{\sqrt{1 + 2\sin x + \sin^2 x}}{\sqrt{\cos^2 x}} = \frac{\sqrt{(1 + \sin x)^2}}{\cos x} \]

### Step 6: Simplify further
Since \( \sqrt{(1 + \sin x)^2} = 1 + \sin x \) (as \( x \) is acute and \( 1 + \sin x \) is positive):
\[ y = \frac{1 + \sin x}{\cos x} \]

### Step 7: Rewrite in terms of trigonometric functions
Now we can express this in terms of secant and tangent:
\[ y = \frac{1}{\cos x} + \frac{\sin x}{\cos x} = \sec x + \tan x \]

### Final Answer
Thus, the simplified expression is:
\[ \sqrt{\frac{1 + \sin x}{1 - \sin x}} = \sec x + \tan x \]