If `sin theta + cos theta = sqrt()3 cos (90- theta)` , then what is the value of tan `theta` /

Options

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

Answer

To solve the equation \( \sin \theta + \cos \theta = \sqrt{3} \cos(90^\circ - \theta) \), we can follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \cos(90^\circ - \theta) = \sin \theta \). Therefore, we can rewrite the equation as:
\[ \sin \theta + \cos \theta = \sqrt{3} \sin \theta \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( \cos \theta \):
\[ \cos \theta = \sqrt{3} \sin \theta - \sin \theta \]
This simplifies to:
\[ \cos \theta = (\sqrt{3} - 1) \sin \theta \]

### Step 3: Divide both sides by \( \cos \theta \)
To find \( \tan \theta \), we can divide both sides of the equation by \( \cos \theta \):
\[ \frac{\cos \theta}{\cos \theta} = \frac{(\sqrt{3} - 1) \sin \theta}{\cos \theta} \]
This simplifies to:
\[ 1 = (\sqrt{3} - 1) \tan \theta \]

### Step 4: Solve for \( \tan \theta \)
Now, we can solve for \( \tan \theta \):
\[ \tan \theta = \frac{1}{\sqrt{3} - 1} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \tan \theta = \frac{1 \cdot (\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{\sqrt{3} + 1}{3 - 1} = \frac{\sqrt{3} + 1}{2} \]

### Conclusion
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{\sqrt{3} + 1}{2} \]