If `2 cot theta =3`, then what is `(2 cos theta -sin theta)/(2 cos theta + sin theta)` equal to ?

Options

A. `2/3`
B. `1/3`
C. `1/2`
D. `3/4`

Answer

To solve the problem, we start with the given equation and proceed step by step.

**Step 1:** Start with the given equation:
\[ 2 \cot \theta = 3 \]

**Step 2:** Solve for \( \cot \theta \):
\[ \cot \theta = \frac{3}{2} \]

**Step 3:** Recall the definition of cotangent:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can express this as:
\[ \frac{\cos \theta}{\sin \theta} = \frac{3}{2} \]

**Step 4:** Cross-multiply to find a relationship between \( \cos \theta \) and \( \sin \theta \):
\[ 2 \cos \theta = 3 \sin \theta \]

**Step 5:** Rearranging gives us:
\[ 2 \cos \theta - 3 \sin \theta = 0 \]
or
\[ 2 \cos \theta = 3 \sin \theta \]

**Step 6:** Now, we need to find the value of:
\[ \frac{2 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta} \]

**Step 7:** Substitute \( 2 \cos \theta = 3 \sin \theta \) into the expression:
\[ \frac{3 \sin \theta - \sin \theta}{3 \sin \theta + \sin \theta} \]

**Step 8:** Simplify the expression:
\[ \frac{(3 - 1) \sin \theta}{(3 + 1) \sin \theta} = \frac{2 \sin \theta}{4 \sin \theta} \]

**Step 9:** Cancel \( \sin \theta \) (assuming \( \sin \theta \neq 0 \)):
\[ \frac{2}{4} = \frac{1}{2} \]

Thus, the final answer is:
\[ \frac{1}{2} \]

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