What is the value of `(4//3) cot^2 (pi//6) + 3 cos^2 (150^@) - 4 cosec^2 45^@ + 8 sin (pi//2)`?

Options

A. 25/4
B. 1
C. -7/2
D. 13/2

Answer

To solve the expression \( \frac{4}{3} \cot^2 \left( \frac{\pi}{6} \right) + 3 \cos^2 (150^\circ) - 4 \csc^2 (45^\circ) + 8 \sin \left( \frac{\pi}{2} \right) \), we will evaluate each trigonometric function step by step.

### Step 1: Calculate \( \cot^2 \left( \frac{\pi}{6} \right) \)
The cotangent function is defined as the reciprocal of the tangent function. We know that:
\[ \cot \left( \frac{\pi}{6} \right) = \frac{1}{\tan \left( \frac{\pi}{6} \right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]
Thus,
\[ \cot^2 \left( \frac{\pi}{6} \right) = (\sqrt{3})^2 = 3 \]

### Step 2: Calculate \( \cos^2 (150^\circ) \)
Using the cosine function:
\[ \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^2(150^\circ) = \left(-\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 3: Calculate \( \csc^2 (45^\circ) \)
The cosecant function is the reciprocal of the sine function:
\[ \csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]
Thus,
\[ \csc^2(45^\circ) = (\sqrt{2})^2 = 2 \]

### Step 4: Calculate \( \sin \left( \frac{\pi}{2} \right) \)
The sine of \( \frac{\pi}{2} \) is:
\[ \sin \left( \frac{\pi}{2} \right) = 1 \]

### Step 5: Substitute the values back into the expression
Now we can substitute these values back into the original expression:
\[ \frac{4}{3} \cdot 3 + 3 \cdot \frac{3}{4} - 4 \cdot 2 + 8 \cdot 1 \]

Calculating each term:
1. \( \frac{4}{3} \cdot 3 = 4 \)
2. \( 3 \cdot \frac{3}{4} = \frac{9}{4} \)
3. \( -4 \cdot 2 = -8 \)
4. \( 8 \cdot 1 = 8 \)

### Step 6: Combine the results
Now combine all these values:
\[ 4 + \frac{9}{4} - 8 + 8 \]
This simplifies to:
\[ 4 + \frac{9}{4} = \frac{16}{4} + \frac{9}{4} = \frac{25}{4} \]
And since \(-8 + 8 = 0\), we have:
\[ \frac{25}{4} \]

### Final Answer
The final value of the expression is:
\[ \frac{25}{4} \]