If tan θ + cot θ = 4, where θ is an acute angle, then find the value of sin θ · cos θ.

The correct answer is $\frac{1}{4}$.

Given:

$\tan \theta +\cot \theta =4$

Step 1: Express $\tan \theta$ and $\cot \theta$ in terms of $\sin \theta$ and $\cos \theta$.

We use the trigonometric identities:

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

$\cot \theta =\frac{\cos \theta }{\sin \theta }$

Substituting these into the given equation:

$\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }=4$

Step 2: Simplify the left-hand side.

Finding a common denominator of $\sin \theta ·\cos \theta$:

$\frac{{\sin }^2 \theta +{\cos }^2 \theta }{\sin \theta ·\cos \theta }=4$

Step 3: Apply the Pythagorean identity.

Using the identity ${\sin }^2 \theta +{\cos }^2 \theta =1$:

$\frac{1}{\sin \theta ·\cos \theta }=4$

Step 4: Solve for $\sin \theta ·\cos \theta$.

Taking the reciprocal of both sides:

$\sin \theta ·\cos \theta =\frac{1}{4}$

Therefore, the value of $\sin \theta ·\cos \theta$ is $\frac{1}{4}$.