If `sintheta.sec(30^(@)+theta)=1` where `(0ltthetalt60^(@))` then the value of `sintheta+cos2theta` is :

Options

A. 1
B. `(2+sqrt(3))/(2sqrt(3))`
C. 0
D. `sqrt(2)`

Answer

To solve the equation \( \sin \theta \cdot \sec(30^\circ + \theta) = 1 \) where \( 0 < \theta < 60^\circ \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \sec(x) = \frac{1}{\cos(x)} \). Therefore, we can rewrite the equation as:
\[ \sin \theta \cdot \frac{1}{\cos(30^\circ + \theta)} = 1 \]
This simplifies to:
\[ \sin \theta = \cos(30^\circ + \theta) \]

### Step 2: Use the cosine addition formula
Using the cosine addition formula, we have:
\[ \cos(30^\circ + \theta) = \cos 30^\circ \cos \theta - \sin 30^\circ \sin \theta \]
Substituting \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \sin 30^\circ = \frac{1}{2} \), we get:
\[ \sin \theta = \frac{\sqrt{3}}{2} \cos \theta - \frac{1}{2} \sin \theta \]

### Step 3: Rearrange the equation
Rearranging the equation gives:
\[ \sin \theta + \frac{1}{2} \sin \theta = \frac{\sqrt{3}}{2} \cos \theta \]
This simplifies to:
\[ \frac{3}{2} \sin \theta = \frac{\sqrt{3}}{2} \cos \theta \]

### Step 4: Divide both sides by \( \cos \theta \)
Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \frac{3}{2} \tan \theta = \frac{\sqrt{3}}{2} \]

### Step 5: Solve for \( \tan \theta \)
This simplifies to:
\[ \tan \theta = \frac{\sqrt{3}}{3} \]
From trigonometric values, we know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \quad \text{or} \quad \tan 60^\circ = \sqrt{3} \]
Thus, \( \theta = 30^\circ \).

### Step 6: Calculate \( \sin \theta + \cos 2\theta \)
Now we need to find \( \sin \theta + \cos 2\theta \):
1. Calculate \( \sin 30^\circ = \frac{1}{2} \).
2. Calculate \( \cos 2\theta = \cos 60^\circ = \frac{1}{2} \).

So,
\[ \sin 30^\circ + \cos 60^\circ = \frac{1}{2} + \frac{1}{2} = 1. \]

### Final Answer
The value of \( \sin \theta + \cos 2\theta \) is \( 1 \).

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