If `A = sin 45^(@)+ cos 45^(@) ` and `B = sin 44^(@) + cos 44^(@)` then

Options

A. `A gt B`
B. ` A lt B`
C. `A=B`
D. none of these

Answer

To solve the problem, we need to find the relationship between \( A \) and \( B \) where:

\[ A = \sin 45^\circ + \cos 45^\circ \]
\[ B = \sin 44^\circ + \cos 44^\circ \]

### Step 1: Calculate \( A \)

First, we calculate \( A \):

\[ A = \sin 45^\circ + \cos 45^\circ \]

Using the values of sine and cosine at \( 45^\circ \):

\[ \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2} \]

Thus,

\[ A = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2} \]

### Step 2: Calculate \( B \)

Next, we calculate \( B \):

\[ B = \sin 44^\circ + \cos 44^\circ \]

### Step 3: Find \( A - B \)

Now, we find \( A - B \):

\[ A - B = \left(\sin 45^\circ + \cos 45^\circ\right) - \left(\sin 44^\circ + \cos 44^\circ\right) \]
\[ = \left(\sin 45^\circ - \sin 44^\circ\right) + \left(\cos 45^\circ - \cos 44^\circ\right) \]

### Step 4: Use Sine and Cosine Difference Formulas

Using the sine and cosine difference formulas:

1. For \( \sin C - \sin D \):
\[ \sin C - \sin D = 2 \cos\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right) \]

2. For \( \cos C - \cos D \):
\[ \cos C - \cos D = -2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right) \]

Applying these formulas:

- Let \( C = 45^\circ \) and \( D = 44^\circ \):

\[ \sin 45^\circ - \sin 44^\circ = 2 \cos\left(\frac{45^\circ + 44^\circ}{2}\right) \sin\left(\frac{45^\circ - 44^\circ}{2}\right) \]
\[ = 2 \cos\left(44.5^\circ\right) \sin\left(0.5^\circ\right) \]

\[ \cos 45^\circ - \cos 44^\circ = -2 \sin\left(\frac{45^\circ + 44^\circ}{2}\right) \sin\left(\frac{45^\circ - 44^\circ}{2}\right) \]
\[ = -2 \sin\left(44.5^\circ\right) \sin\left(0.5^\circ\right) \]

### Step 5: Combine the Results

Combining these results:

\[ A - B = \left(2 \cos(44.5^\circ) \sin(0.5^\circ) - 2 \sin(44.5^\circ) \sin(0.5^\circ)\right) \]
\[ = 2 \sin(0.5^\circ) \left(\cos(44.5^\circ) - \sin(44.5^\circ)\right) \]

### Step 6: Analyze the Sign of \( A - B \)

Since \( \sin(0.5^\circ) > 0 \), the sign of \( A - B \) depends on \( \cos(44.5^\circ) - \sin(44.5^\circ) \).

### Conclusion

For \( 0 < \theta < \frac{\pi}{4} \), \( \cos \theta > \sin \theta \), hence:

\[ A - B > 0 \implies A > B \]

Thus, we conclude that \( A \) is greater than \( B \).