If `"tan"(pi)/(3)+"cosec"(pi)/(6)=x` , then x is

Options

A. `((1-sqrt(2)))/(sqrt(2))`
B. `((sqrt(3)+4))/(2sqrt(3))`
C. `sqrt(3)+2`
D. `((2sqrt(2)-1))/(2)`

Answer

To solve the equation \( \tan\left(\frac{\pi}{3}\right) + \csc\left(\frac{\pi}{6}\right) = x \), we will calculate each trigonometric function step by step.

### Step 1: Calculate \( \tan\left(\frac{\pi}{3}\right) \)

The angle \( \frac{\pi}{3} \) radians corresponds to \( 60^\circ \). The tangent of \( 60^\circ \) is known to be:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 2: Calculate \( \csc\left(\frac{\pi}{6}\right) \)

The angle \( \frac{\pi}{6} \) radians corresponds to \( 30^\circ \). The cosecant is the reciprocal of the sine function. Therefore:
\[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus:
\[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2 \]

### Step 3: Combine the results

Now we can substitute the values we calculated into the equation:
\[ x = \tan\left(\frac{\pi}{3}\right) + \csc\left(\frac{\pi}{6}\right) = \sqrt{3} + 2 \]

### Final Answer

Thus, the value of \( x \) is:
\[ x = \sqrt{3} + 2 \]