Question

The value of \(\frac{sin^23A}{sin^2A}-\frac{cos^23A}{cos^2A}\) is

• A. \(cos2A\)
• B. \(8cos2A\)
• C. \(\frac{1}{8}cos2A\)
• D. \(\frac{1}{2}sin2A\)

Hint:

Use sin3A/sinA = 3 - 4 sin^2 A and cos3A/cosA = 4 cos^2 A - 3.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Triple angle formulas: \(\sin 3A = 3\sin A - 4\sin^3 A\) and \(\cos 3A = 4\cos^3 A - 3\cos A\).

Step 2: Rewrite the ratios:
\[ \frac{\sin 3A}{\sin A} = 3 - 4\sin^2 A, \qquad \frac{\cos 3A}{\cos A} = 4\cos^2 A - 3 \]

Step 3: Use difference of squares:
The expression is \((3 - 4\sin^2A)^2 - (4\cos^2A - 3)^2\).
\[ = \big[(3 - 4\sin^2A) - (4\cos^2A - 3)\big]\big[(3 - 4\sin^2A) + (4\cos^2A - 3)\big] \]
First bracket: \(6 - 4(\sin^2A + \cos^2A) = 2\).
Second bracket: \(4(\cos^2A - \sin^2A) = 4\cos 2A\).

Step 4: Result:
\[ 2 \times 4\cos 2A = 8\cos 2A \]
Options (A), (C) and (D) are off by a constant factor or use \(\sin 2A\).

Final Answer:
The expression simplifies to 8 cos 2A. \[ \boxed{\text{(B) }8\cos 2A} \]