Question

If \(\theta\) is any angle, then \[ \sin^2\theta \cos^2\theta= \]

• A. \(1-\cos 2\theta\)
• B. \(1-\cos 4\theta\)
• C. \(\dfrac{1}{4}(1-\cos 4\theta)\)
• D. \(\dfrac{1}{8}(1-\cos 4\theta)\)

Hint:

For expressions involving \(\sin^2\theta\cos^2\theta\), first use \[ \sin 2\theta=2\sin\theta\cos\theta \] and then apply the power reduction identity.

Answer 1

The Correct Option is D

Step 1: Start with the given expression.
We need to simplify \[ \sin^2\theta \cos^2\theta \]

Step 2: Use the double angle identity.
We know that \[ \sin 2\theta=2\sin\theta\cos\theta \]

Step 3: Square both sides.
Squaring, \[ \sin^2 2\theta=4\sin^2\theta\cos^2\theta \] Therefore, \[ \sin^2\theta\cos^2\theta=\frac{1}{4}\sin^2 2\theta \]

Step 4: Use the power reduction identity.
We know that \[ \sin^2 A=\frac{1-\cos 2A}{2} \] Putting \[ A=2\theta, \] we get \[ \sin^2 2\theta=\frac{1-\cos 4\theta}{2} \]

Step 5: Substitute this value.
Now, \[ \sin^2\theta\cos^2\theta = \frac{1}{4}\cdot \frac{1-\cos 4\theta}{2} \] \[ = \frac{1}{8}(1-\cos 4\theta) \]

Step 6: Match with the options.
The obtained expression is \[ \frac{1}{8}(1-\cos 4\theta) \]

Step 7: Final conclusion.
Therefore, \[ \boxed{\frac{1}{8}(1-\cos 4\theta)} \]