The value of `(cos 0^(@) + sin 45^(@) + sin 30^(@)) (sin 90^(@) - cos 45^(@) + cos 60^(@))` is

Options

A. 0
B. 1
C. `(7)/(4)`
D. `(1)/(2)`

Answer

To solve the expression `(cos 0° + sin 45° + sin 30°) (sin 90° - cos 45° + cos 60°)`, we will first evaluate each trigonometric function using standard angle values.

### Step-by-Step Solution:

1. **Evaluate each trigonometric function:**
- \( \cos 0° = 1 \)
- \( \sin 45° = \frac{1}{\sqrt{2}} \)
- \( \sin 30° = \frac{1}{2} \)
- \( \sin 90° = 1 \)
- \( \cos 45° = \frac{1}{\sqrt{2}} \)
- \( \cos 60° = \frac{1}{2} \)

2. **Substitute the values into the expression:**
\[ (1 + \frac{1}{\sqrt{2}} + \frac{1}{2}) (1 - \frac{1}{\sqrt{2}} + \frac{1}{2}) \]

3. **Simplify the first bracket:**
- Combine \( 1 + \frac{1}{2} = \frac{3}{2} \)
- Thus, the first bracket becomes:
\[ \frac{3}{2} + \frac{1}{\sqrt{2}} \]

4. **Simplify the second bracket:**
- Combine \( 1 + \frac{1}{2} = \frac{3}{2} \)
- Thus, the second bracket becomes:
\[ \frac{3}{2} - \frac{1}{\sqrt{2}} \]

5. **Now, we have:**
\[ \left(\frac{3}{2} + \frac{1}{\sqrt{2}}\right) \left(\frac{3}{2} - \frac{1}{\sqrt{2}}\right) \]

6. **Recognize this as a difference of squares:**
- The expression can be simplified using the formula \( (a + b)(a - b) = a^2 - b^2 \).
- Here, \( a = \frac{3}{2} \) and \( b = \frac{1}{\sqrt{2}} \).

7. **Calculate \( a^2 \) and \( b^2 \):**
- \( a^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \)
- \( b^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \)

8. **Subtract \( b^2 \) from \( a^2 \):**
\[ \frac{9}{4} - \frac{1}{2} = \frac{9}{4} - \frac{2}{4} = \frac{7}{4} \]

9. **Final answer:**
\[ \text{The value of the expression is } \frac{7}{4}. \]