What is the value of `cot 45^(@) +cosec 60^(@)?`

Options

A. `((sqrt6+1)/sqrt3)`
B. `((1+sqrt3)/(2))`
C. `(5)/sqrt3`
D. `((sqrt3+2))/sqrt3`

Answer

To solve the problem, we need to find the value of \( \cot 45^\circ + \csc 60^\circ \).

### Step-by-step Solution:

1. **Calculate \( \cot 45^\circ \)**:
- The cotangent of an angle is the reciprocal of the tangent.
- We know that \( \tan 45^\circ = 1 \).
- Therefore, \( \cot 45^\circ = \frac{1}{\tan 45^\circ} = \frac{1}{1} = 1 \).

2. **Calculate \( \csc 60^\circ \)**:
- The cosecant of an angle is the reciprocal of the sine.
- We know that \( \sin 60^\circ = \frac{\sqrt{3}}{2} \).
- Therefore, \( \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).

3. **Add the two values**:
- Now we add \( \cot 45^\circ \) and \( \csc 60^\circ \):
\[ \cot 45^\circ + \csc 60^\circ = 1 + \frac{2}{\sqrt{3}}. \]

4. **Combine the terms**:
- To combine these terms, we need a common denominator. The common denominator here is \( \sqrt{3} \).
- Rewrite \( 1 \) as \( \frac{\sqrt{3}}{\sqrt{3}} \):
\[ 1 + \frac{2}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{\sqrt{3} + 2}{\sqrt{3}}. \]

5. **Final Result**:
- Therefore, the final value of \( \cot 45^\circ + \csc 60^\circ \) is:
\[ \frac{\sqrt{3} + 2}{\sqrt{3}}. \]