Question

Simplest form of \(\frac{\sec A}{\sqrt{\sec^2 A - 1}}\) is

• A. \(\sin A\)
• B. \(\tan A\)
• C. \(\csc A\)
• D. \(\cos A\)

Hint:

Whenever you see \(\sqrt{\sec^2 A - 1}\) or \(\sqrt{1 - \sin^2 A}\), immediately replace them using basic identities (\(\tan A\) and \(\cos A\) respectively) to simplify.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question involves using fundamental trigonometric identities to simplify an expression.
Step 2: Key Formula or Approach:
Recall the identity: \(1 + \tan^2 A = \sec^2 A \implies \sec^2 A - 1 = \tan^2 A\).
Step 3: Detailed Explanation:
The given expression is:
\[ \frac{\sec A}{\sqrt{\sec^2 A - 1}} \]
Substitute \(\sec^2 A - 1 = \tan^2 A\):
\[ = \frac{\sec A}{\sqrt{\tan^2 A}} \]
\[ = \frac{\sec A}{\tan A} \]
Expressing in terms of \(\sin A\) and \(\cos A\):
\[ = \frac{\frac{1}{\cos A}}{\frac{\sin A}{\cos A}} \]
\[ = \frac{1}{\cos A} \times \frac{\cos A}{\sin A} \]
\[ = \frac{1}{\sin A} = \csc A \]
Step 4: Final Answer:
The simplest form is \(\csc A\).