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:

Converting all trigonometric terms to sine and cosine is a reliable, error-free strategy for simplifying expressions.
Always recall your identity relations: $\sec^2 A - 1 = \tan^2 A$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a trigonometric expression:
\[ \frac{\sec A}{\sqrt{\sec^2 A - 1}} \]
We need to simplify this expression to its basic form.

Step 2: Key Formula or Approach:
We will use the fundamental trigonometric Pythagorean identities:
\[ \sec^2 A - \tan^2 A = 1 \implies \sec^2 A - 1 = \tan^2 A \]
Hence:
\[ \sqrt{\sec^2 A - 1} = \tan A \]
We also express $\sec A$ and $\tan A$ in terms of $\sin A$ and $\cos A$ to simplify the fraction.

Step 3: Detailed Explanation:

• Write the given expression:
\[ \text{Expression} = \frac{\sec A}{\sqrt{\sec^2 A - 1}} \]

• Substitute the identity $\sqrt{\sec^2 A - 1} = \tan A$ into the denominator:
\[ \text{Expression} = \frac{\sec A}{\tan A} \]

• Express $\sec A$ and $\tan A$ in terms of sine and cosine:
\[ \sec A = \frac{1}{\cos A} \]
\[ \tan A = \frac{\sin A}{\cos A} \]

• Substitute these into the simplified ratio:
\[ \text{Expression} = \frac{\frac{1}{\cos A}}{\frac{\sin A}{\cos A}} \]

• Simplify the compound fraction:
\[ \text{Expression} = \frac{1}{\cos A} \times \frac{\cos A}{\sin A} \]
\[ \text{Expression} = \frac{1}{\sin A} \]

• Use the reciprocal identity $\frac{1}{\sin A} = \csc A$:
\[ \text{Expression} = \csc A \]


Step 4: Final Answer:
The simplest form of the given expression is $\csc A$.