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: Understanding the Question:
The topic of this question is Trigonometric Identities.
We are asked to simplify a given trigonometric ratio to find its equivalent form among the choices.

Step 2: Key Formula or Approach:
Recall the fundamental Pythagorean identity relating secant and tangent:
\[ \sec^2 A - 1 = \tan^2 A \] Also recall the quotient definition of the tangent function:
\[ \tan A = \frac{\sin A}{\cos A} \implies \tan^2 A = \frac{\sin^2 A}{\cos^2 A} \] And the reciprocal relationship for secant:
\[ \sec^2 A = \frac{1}{\cos^2 A} \]

Step 3: Detailed Explanation:

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

• Substitute the Pythagorean identity \(\sec^2 A - 1 = \tan^2 A\) into the numerator:
\[ \text{Expression} = \frac{\tan^2 A}{\sin^2 A} \]

• Write \(\tan^2 A\) in terms of sine and cosine:
\[ \text{Expression} = \frac{\frac{\sin^2 A}{\cos^2 A}}{\sin^2 A} \]

• Simplify the complex fraction:
Multiply the numerator by the reciprocal of the denominator:
\[ \text{Expression} = \frac{\sin^2 A}{\cos^2 A} \times \frac{1}{\sin^2 A} \] The \(\sin^2 A\) terms cancel out:
\[ \text{Expression} = \frac{1}{\cos^2 A} \]

• Convert this term back to the secant function:
\[ \frac{1}{\cos^2 A} = \sec^2 A \]

Step 4: Final Answer:
The expression \(\frac{\sec^2 A - 1}{\sin^2 A}\) is equal to \(\sec^2 A\).
Therefore, the correct option is (B).