If \(sin^2 A - cos^2 A = 1/4\) , then find the value of \(cos^2 A\) .

Question

If \(sin^2 A - cos^2 A = 1/4\) , then find the value of \(cos^2 A\) .

1. A. \(1/4\)
2. B. \(3/8\) (Correct answer)
3. C. \(5/8\)
4. D. \(1/2\)

Correct Answer

Option B — \(3/8\)

Detailed Solution & Explanation

The correct answer is 3/8.

Key Points

• Use the fundamental identity sin²A + cos²A = 1, so sin²A = 1 − cos²A.
• Substitute into the given equation:
  • (1 − cos²A) − cos²A = 1/4
  • 1 − 2cos²A = 1/4
  • 2cos²A = 3/4 → cos²A = 3/8

Additional Information

• Check: cos²A = 3/8 gives sin²A = 5/8, and 5/8 − 3/8 = 2/8 = 1/4 ✓.
• The three Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ.
• Note also that sin²A − cos²A = −cos 2A, so the equation says cos 2A = −1/4 — an alternative route via the double-angle formula cos 2A = 2cos²A − 1, which gives the same result in one step.
• Expressing everything in terms of a single trigonometric ratio is the standard first move; here replacing sin²A removed the second unknown in one step.
• The complementary result follows immediately: sin²A = 1 − 3/8 = 5/8, so tan²A = sin²A/cos²A = (5/8)/(3/8) = 5/3.

प्रश्न (हिन्दी में)

यदि \(sin^2 A - cos^2 A = 1/4\) हो, तो \(cos^2 A\) का मान ज्ञात कीजिए।

1. A. \(1/4\)
2. B. \(3/8\)
3. C. \(5/8\)
4. D. \(1/2\)