If `thetaand2theta-45^(@)` are acute angles such that `sintheta=cos(2theta-45^(@))` then `tantheta` is

Options

A. 1
B. -1
C. `sqrt3`
D. `1/sqrt3`

Answer

To solve the problem, we need to find the value of \( \tan \theta \) given that \( \sin \theta = \cos(2\theta - 45^\circ) \) and both \( \theta \) and \( 2\theta - 45^\circ \) are acute angles.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that \( \sin \theta = \cos(2\theta - 45^\circ) \). Since \( \sin \theta \) can be expressed in terms of cosine, we can use the complementary angle identity:
\[ \sin \theta = \cos(90^\circ - \theta) \]
This means:
\[ \cos(2\theta - 45^\circ) = \cos(90^\circ - \theta) \]

2. **Setting the Angles Equal**:
Since the cosine function is equal, we can set the angles equal to each other:
\[ 2\theta - 45^\circ = 90^\circ - \theta \]

3. **Solving for \( \theta \)**:
Rearranging the equation:
\[ 2\theta + \theta = 90^\circ + 45^\circ \]
\[ 3\theta = 135^\circ \]
Dividing both sides by 3:
\[ \theta = 45^\circ \]

4. **Finding \( \tan \theta \)**:
Now that we have \( \theta = 45^\circ \), we can find \( \tan \theta \):
\[ \tan 45^\circ = 1 \]

### Conclusion:
Thus, the value of \( \tan \theta \) is \( 1 \).