If `sqrt(3)sectheta- 2tantheta=0` and `0^(@) lt theta lt 90^(@)` find the value of `theta`

Answer

To solve the equation \( \sqrt{3} \sec \theta - 2 \tan \theta = 0 \) for \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Rearrange the equation
Start by isolating one of the trigonometric functions:
\[ \sqrt{3} \sec \theta = 2 \tan \theta \]

### Step 2: Substitute trigonometric identities
Recall the definitions of secant and tangent:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Substituting these into the equation gives:
\[ \sqrt{3} \cdot \frac{1}{\cos \theta} = 2 \cdot \frac{\sin \theta}{\cos \theta} \]

### Step 3: Simplify the equation
Since \( \cos \theta \) is not zero in the interval \( (0^\circ, 90^\circ) \), we can multiply both sides by \( \cos \theta \):
\[ \sqrt{3} = 2 \sin \theta \]

### Step 4: Solve for \( \sin \theta \)
Now, isolate \( \sin \theta \):
\[ \sin \theta = \frac{\sqrt{3}}{2} \]

### Step 5: Find \( \theta \)
We know from trigonometric values that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]
Thus, we conclude:
\[ \theta = 60^\circ \]

### Final Answer
The value of \( \theta \) is \( 60^\circ \).

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