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: Square the whole expression first.
Instead of substituting sine and cosine right away, square the given expression: $\left(\frac{\sec A}{\sqrt{\sec^2 A - 1}}\right)^2 = \frac{\sec^2 A}{\sec^2 A - 1}$.
Step 2: Use the identity to replace the denominator.
Since $\sec^2 A - 1 = \tan^2 A$, this becomes $\frac{\sec^2 A}{\tan^2 A} = \frac{1/\cos^2 A}{\sin^2 A/\cos^2 A} = \frac{1}{\sin^2 A} = \csc^2 A$.
Step 3: Take the square root to return to the original expression.
Since $A$ is an angle in a right triangle, all ratios here are positive, so taking the positive square root of both sides gives the original expression equal to $\csc A$.
\[ \boxed{\csc A} \]