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: Recognise the product of sin and cos.
We want to simplify $\sin^2\theta \cos^2\theta$. The key insight is to connect this to a double-angle formula. Notice that $\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta$.
Step 2: Square the double angle relation.
Squaring $\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta$ gives \[ \sin^2\theta\cos^2\theta = \frac{1}{4}\sin^2 2\theta. \]
Step 3: Apply the power reduction formula.
We use the identity $\sin^2 A = \frac{1 - \cos 2A}{2}$. Setting $A = 2\theta$: \[ \sin^2 2\theta = \frac{1 - \cos 4\theta}{2}. \]
Step 4: Substitute back.
\[ \sin^2\theta\cos^2\theta = \frac{1}{4} \cdot \frac{1-\cos 4\theta}{2} = \frac{1-\cos 4\theta}{8}. \]
Step 5: Match to the options.
The result $\frac{1}{8}(1-\cos 4\theta)$ matches option (4).
Step 6: State the final answer.
\[ \boxed{\dfrac{1}{8}(1-\cos 4\theta)} \]