Question

\(\frac{\sec^2 A - 1}{\sin^2 A}\) is same as

• A. \(\cos^2 A\)
• B. \(\sec^2 A\)
• C. \(-\sec^2 A\)
• D. \(\cot^2 A\)

Hint:

Using basic identities like \(\tan A = \frac{\sin A}{\cos A}\) can simplify complex-looking fractions instantly.
Always substitute the numerator identity first to see if any terms cancel!

Answer 1

The Correct Option is B

Step 1: Convert everything to sine and cosine right away.
$\sec^2A-1=\frac{1}{\cos^2A}-1=\frac{1-\cos^2A}{\cos^2A}=\frac{\sin^2A}{\cos^2A}$, using $1-\cos^2A=\sin^2A$.
Step 2: Divide by $\sin^2A$.
$\frac{\sec^2A-1}{\sin^2A}=\frac{\sin^2A}{\cos^2A\cdot\sin^2A}=\frac{1}{\cos^2A}$.
Step 3: Rewrite as secant.
$\frac{1}{\cos^2A}=\sec^2A$, matching option (B).
\[ \boxed{\sec^2 A} \]