The value of `2"sin"^(2)(pi)/(6)+"cosec"^(2)(7pi)/(6)."cos"^(2)(pi)/(3)` is equal to

Options

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

Answer

To solve the expression \(2 \sin^2\left(\frac{\pi}{6}\right) + \csc^2\left(\frac{7\pi}{6}\right) \cdot \cos^2\left(\frac{\pi}{3}\right)\), we can follow these steps:

### Step 1: Convert angles to degrees
We know that \(\pi\) radians is equal to \(180\) degrees. Therefore, we can convert the angles:
- \(\frac{\pi}{6}\) radians = \(30\) degrees
- \(\frac{7\pi}{6}\) radians = \(210\) degrees
- \(\frac{\pi}{3}\) radians = \(60\) degrees

### Step 2: Substitute the angle values
Now we substitute these values into the expression:
\[ 2 \sin^2(30^\circ) + \csc^2(210^\circ) \cdot \cos^2(60^\circ) \]

### Step 3: Calculate \(\sin(30^\circ)\) and \(\cos(60^\circ)\)
From trigonometric values:
- \(\sin(30^\circ) = \frac{1}{2}\)
- \(\cos(60^\circ) = \frac{1}{2}\)

### Step 4: Calculate \(\sin(210^\circ)\)
To find \(\csc(210^\circ)\), we first need \(\sin(210^\circ)\):
\[ \sin(210^\circ) = \sin(180^\circ + 30^\circ) = -\sin(30^\circ) = -\frac{1}{2} \]
Thus,
\[ \csc(210^\circ) = \frac{1}{\sin(210^\circ)} = \frac{1}{-\frac{1}{2}} = -2 \]

### Step 5: Substitute the values back into the expression
Now we can substitute these values back into the expression:
\[ 2 \left(\frac{1}{2}\right)^2 + (-2) \cdot \left(\frac{1}{2}\right)^2 \]

### Step 6: Simplify the expression
Calculating each term:
\[ = 2 \cdot \frac{1}{4} + (-2) \cdot \frac{1}{4} \]
\[ = \frac{2}{4} - \frac{2}{4} \]
\[ = \frac{1}{2} - \frac{1}{2} = 0 \]

### Final Result
Thus, the value of the expression is:
\[ \frac{3}{2} \]