If `tan3theta=cot(75^(@)-2theta)` then find the value of `sin 4 theta`

Answer

To solve the equation \( \tan 3\theta = \cot(75^\circ - 2\theta) \) and find the value of \( \sin 4\theta \), we can follow these steps:

### Step 1: Set up the equation
Given the relationship \( \tan x = \cot y \), we know that \( x + y = 90^\circ \). In our case, we have:
\[ 3\theta + (75^\circ - 2\theta) = 90^\circ \]

### Step 2: Simplify the equation
Now, we can simplify the equation:
\[ 3\theta + 75^\circ - 2\theta = 90^\circ \]
Combine like terms:
\[ \theta + 75^\circ = 90^\circ \]

### Step 3: Solve for \( \theta \)
Subtract \( 75^\circ \) from both sides:
\[ \theta = 90^\circ - 75^\circ \]
\[ \theta = 15^\circ \]

### Step 4: Find \( 4\theta \)
Now that we have \( \theta \), we can find \( 4\theta \):
\[ 4\theta = 4 \times 15^\circ = 60^\circ \]

### Step 5: Calculate \( \sin 4\theta \)
Now we need to find \( \sin 4\theta \):
\[ \sin 4\theta = \sin 60^\circ \]
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( \sin 4\theta \) is:
\[ \sin 4\theta = \frac{\sqrt{3}}{2} \]

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