The value of sin `0^(@) + cos 30^(@) - tan 45^(@) + cosec 60^(@) + cot 90^(@) ` is equal to

Options

A. `(5 sqrt(3) - 6)/(6)`
B. `(-6 + 7 sqrt(3))/(6)`
C. 0
D. 2

Answer

To solve the expression \( \sin 0^\circ + \cos 30^\circ - \tan 45^\circ + \csc 60^\circ + \cot 90^\circ \), we will evaluate each trigonometric function step by step.

### Step 1: Evaluate \( \sin 0^\circ \)
\[ \sin 0^\circ = 0 \]

### Step 2: Evaluate \( \cos 30^\circ \)
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

### Step 3: Evaluate \( \tan 45^\circ \)
\[ \tan 45^\circ = 1 \]

### Step 4: Evaluate \( \csc 60^\circ \)
\[ \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 5: Evaluate \( \cot 90^\circ \)
\[ \cot 90^\circ = 0 \]

### Step 6: Substitute the values into the expression
Now we substitute the values we found into the original expression:
\[ 0 + \frac{\sqrt{3}}{2} - 1 + \frac{2}{\sqrt{3}} + 0 \]
This simplifies to:
\[ \frac{\sqrt{3}}{2} - 1 + \frac{2}{\sqrt{3}} \]

### Step 7: Combine the terms
To combine the terms, we need a common denominator. The least common multiple of the denominators \(2\) and \(\sqrt{3}\) is \(2\sqrt{3}\).

Rewriting each term with the common denominator:
- The first term:
\[ \frac{\sqrt{3}}{2} = \frac{\sqrt{3} \cdot \sqrt{3}}{2 \cdot \sqrt{3}} = \frac{3}{2\sqrt{3}} \]
- The second term:
\[ -1 = -\frac{2\sqrt{3}}{2\sqrt{3}} = -\frac{2\sqrt{3}}{2\sqrt{3}} \]
- The third term:
\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot 2}{\sqrt{3} \cdot 2} = \frac{4}{2\sqrt{3}} \]

Now we can combine:
\[ \frac{3}{2\sqrt{3}} - \frac{2\sqrt{3}}{2\sqrt{3}} + \frac{4}{2\sqrt{3}} = \frac{3 - 2\sqrt{3} + 4}{2\sqrt{3}} = \frac{7 - 2\sqrt{3}}{2\sqrt{3}} \]

### Step 8: Rationalize the denominator
To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{3}\):
\[ \frac{(7 - 2\sqrt{3})\sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{7\sqrt{3} - 6}{6} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{7\sqrt{3} - 6}{6} \]