If `A=sin 44^(@)+cos44^(@), B=sin45^(@)+ cos45^(@)` and `C= sin46^(@)+cos 46^(@)`. Then, correct option(s) is/are

Options

A. `A lt B lt C`
B. `C lt B lt A`
C. `B gt A`
D. `A=C`

Answer

To solve the problem, we need to evaluate the expressions for \( A \), \( B \), and \( C \) and compare their values.

### Step 1: Calculate \( A = \sin 44^\circ + \cos 44^\circ \)

Using the identity \( \sin x + \cos x = \sqrt{2} \sin\left(x + 45^\circ\right) \):

\[ A = \sin 44^\circ + \cos 44^\circ = \sqrt{2} \sin\left(44^\circ + 45^\circ\right) = \sqrt{2} \sin 89^\circ \]

Since \( \sin 89^\circ \approx 1 \):

\[ A \approx \sqrt{2} \cdot 1 = \sqrt{2} \]

### Step 2: Calculate \( B = \sin 45^\circ + \cos 45^\circ \)

Using the known values:

\[ B = \sin 45^\circ + \cos 45^\circ = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 3: Calculate \( C = \sin 46^\circ + \cos 46^\circ \)

Using the same identity:

\[ C = \sin 46^\circ + \cos 46^\circ = \sqrt{2} \sin\left(46^\circ + 45^\circ\right) = \sqrt{2} \sin 91^\circ \]

Since \( \sin 91^\circ \approx 1 \):

\[ C \approx \sqrt{2} \cdot 1 = \sqrt{2} \]

### Step 4: Compare \( A \), \( B \), and \( C \)

From the calculations:

\[ A \approx \sqrt{2}, \quad B = \sqrt{2}, \quad C \approx \sqrt{2} \]

Thus, we find that:

\[ A = B = C \]

### Conclusion

All three values are equal, hence the correct option(s) is/are:

- \( A = B = C \)