What is the simplified value of
`sqrt((secA-1)/(secA+1))`?

Options

A. `"cosec"A-cotA`
B. `secA-tanA`
C. `sec^(2)A`
D. `secA"cosec"A`

Answer

To simplify the expression \( \sqrt{\frac{\sec A - 1}{\sec A + 1}} \), we can follow these steps:

### Step 1: Rewrite secant in terms of cosine
Recall that \( \sec A = \frac{1}{\cos A} \). Therefore, we can rewrite the expression as:
\[ \sqrt{\frac{\frac{1}{\cos A} - 1}{\frac{1}{\cos A} + 1}} \]

### Step 2: Simplify the numerator and denominator
Now, we simplify the numerator and denominator separately:
- The numerator becomes:
\[ \frac{1 - \cos A}{\cos A} \]
- The denominator becomes:
\[ \frac{1 + \cos A}{\cos A} \]
Thus, the expression now looks like:
\[ \sqrt{\frac{\frac{1 - \cos A}{\cos A}}{\frac{1 + \cos A}{\cos A}}} \]

### Step 3: Simplify the fraction
Since both the numerator and denominator have the same denominator \( \cos A \), we can cancel it out:
\[ \sqrt{\frac{1 - \cos A}{1 + \cos A}} \]

### Step 4: Rationalize the expression
To simplify \( \sqrt{\frac{1 - \cos A}{1 + \cos A}} \), we can multiply the numerator and denominator by \( 1 - \cos A \):
\[ \sqrt{\frac{(1 - \cos A)^2}{(1 + \cos A)(1 - \cos A)}} \]
This simplifies to:
\[ \sqrt{\frac{(1 - \cos A)^2}{1 - \cos^2 A}} = \sqrt{\frac{(1 - \cos A)^2}{\sin^2 A}} \]

### Step 5: Take the square root
Taking the square root gives:
\[ \frac{1 - \cos A}{\sin A} \]

### Step 6: Use trigonometric identities
We know that:
\[ \frac{1 - \cos A}{\sin A} = \tan\left(\frac{A}{2}\right) \]
This is derived from the half-angle identities.

### Final Result
Thus, the simplified value of \( \sqrt{\frac{\sec A - 1}{\sec A + 1}} \) is:
\[ \tan\left(\frac{A}{2}\right) \]

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