If `sintheta+costheta=sqrt(3)cos(90-theta)`, what is the value of `tantheta` ?

Options

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

Answer

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

### Step 1: Rewrite the equation
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 terms involving \( \sin \theta \):
\[ \sin \theta + \cos \theta - \sqrt{3} \sin \theta = 0 \]
This simplifies to:
\[ (1 - \sqrt{3}) \sin \theta + \cos \theta = 0 \]

### Step 3: Isolate \( \sin \theta \)
Next, we can isolate \( \sin \theta \):
\[ (1 - \sqrt{3}) \sin \theta = -\cos \theta \]

### Step 4: Divide by \( \cos \theta \)
To find \( \tan \theta \), we can divide the entire equation by \( \cos \theta \):
\[ \frac{(1 - \sqrt{3}) \sin \theta}{\cos \theta} = -1 \]
This gives us:
\[ (1 - \sqrt{3}) \tan \theta = -1 \]

### Step 5: Solve for \( \tan \theta \)
Now, we can solve for \( \tan \theta \):
\[ \tan \theta = \frac{-1}{1 - \sqrt{3}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate \( 1 + \sqrt{3} \):
\[ \tan \theta = \frac{-1(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
The denominator simplifies as follows:
\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \]
Thus, we have:
\[ \tan \theta = \frac{-(1 + \sqrt{3})}{-2} = \frac{1 + \sqrt{3}}{2} \]

### Final Answer
The value of \( \tan \theta \) is:
\[ \tan \theta = \frac{1 + \sqrt{3}}{2} \]

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