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: Test the expression at a convenient specific angle instead of simplifying algebraically.
Let $A = 45^\circ$. Then $\sec A = \sqrt{2}$, so $\sec^2A - 1 = 2-1=1$, and $\sqrt{\sec^2A-1}=1$.
Step 2: Evaluate the given expression at this angle. \[ \frac{\sec A}{\sqrt{\sec^2A-1}} = \frac{\sqrt{2}}{1} = \sqrt{2} \]
Step 3: Check which option equals sqrt(2) at A = 45 degrees.
$\sin 45^\circ = \frac{1}{\sqrt2}$, $\tan45^\circ=1$, $\cos45^\circ=\frac{1}{\sqrt2}$, but $\csc45^\circ = \sqrt2$. Only $\csc A$ matches.
\[ \boxed{\csc A} \]