If $\frac{3\sin \theta -\cos \theta }{\cos \theta +\sin \theta }=1$, then the value of $\cot \theta$ is:

The correct answer is 1.

We are given the equation:

$\frac{3\sin \theta -\cos \theta }{\cos \theta +\sin \theta }=1$

and we need to find the value of $\cot \theta$.

Step 1: Cross-multiply to eliminate the fraction.

Since the right-hand side is 1, multiplying both sides by $(\cos \theta +\sin \theta )$ gives:

$3\sin \theta -\cos \theta =\cos \theta +\sin \theta$

Step 2: Rearrange by grouping the sinθ terms and cosθ terms.

Bring all $\sin \theta$ terms to the left and all $\cos \theta$ terms to the right:

$3\sin \theta -\sin \theta =\cos \theta +\cos \theta$

$2\sin \theta =2\cos \theta$

Step 3: Simplify.

Divide both sides by 2:

$\sin \theta =\cos \theta$

Step 4: Find cotθ.

Divide both sides by $\sin \theta$:

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

By definition, $\cot \theta =\frac{\cos \theta }{\sin \theta }$, so:

$\cot \theta =1$

Verification: When $\cot \theta =1$, we have $\theta =45°$, so $\sin \theta =\cos \theta =\frac{1}{\sqrt{2}}$. Substituting back:
$\frac{3·\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=\frac{\frac{2}{\sqrt{2}}}{\frac{2}{\sqrt{2}}}=1$ ✓

Therefore, the value of $\cot \theta$ is 1.