If `(secA)/(sqrt((sec^(2)A-1)))=x` then the value of x is

Options

A. tan A
B. cosec A
C. sin A
D. cos A

Answer

To solve the equation \(\frac{\sec A}{\sqrt{\sec^2 A - 1}} = x\), we will follow these steps:

### Step 1: Recall the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 A - 1 = \tan^2 A \]
This means we can rewrite the expression under the square root.

### Step 2: Substitute the Identity
Substituting the identity into the equation gives us:
\[ \sqrt{\sec^2 A - 1} = \sqrt{\tan^2 A} = \tan A \]
Thus, the equation becomes:
\[ \frac{\sec A}{\tan A} = x \]

### Step 3: Rewrite Secant and Tangent
We can express \(\sec A\) and \(\tan A\) in terms of sine and cosine:
\[ \sec A = \frac{1}{\cos A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]
Substituting these into the equation gives:
\[ \frac{\frac{1}{\cos A}}{\frac{\sin A}{\cos A}} = x \]

### Step 4: Simplify the Expression
The \(\cos A\) in the numerator and denominator cancels out:
\[ \frac{1}{\sin A} = x \]

### Step 5: Final Result
Thus, we find that:
\[ x = \csc A \]

### Conclusion
The value of \(x\) is \(\csc A\).

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