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: Try a specific value of $A$ instead of general algebra.
Let $A = 45^\circ$, so $\sec A = \sqrt{2}$.
Step 2: Plug into the expression.
\[ \frac{\sec A}{\sqrt{\sec^2 A - 1}} = \frac{\sqrt{2}}{\sqrt{2-1}} = \frac{\sqrt{2}}{1} = \sqrt{2} \]
Step 3: Compare with each option evaluated at $A = 45^\circ$.
$\sin 45^\circ = \frac{1}{\sqrt2}$, $\tan 45^\circ = 1$, $\csc 45^\circ = \sqrt2$, $\cos 45^\circ = \frac{1}{\sqrt2}$. Only $\csc A$ gives $\sqrt2$, matching our computed value.
Step 4: Conclude.
The simplest form of the expression is $\csc A$, option (3).
\[ \boxed{\csc A} \]