If `0^(@)lt theta lt 90^(@)` and `cos theta=(4)/(5)` find the values of
`cos(90^(@)+theta)`

Answer

To solve the problem, we need to find the value of \( \cos(90^\circ + \theta) \) given that \( \cos \theta = \frac{4}{5} \) and \( 0^\circ < \theta < 90^\circ \).

### Step-by-Step Solution:

1. **Use the Cosine Addition Formula**:
We know that:
\[ \cos(90^\circ + \theta) = \cos 90^\circ \cos \theta - \sin 90^\circ \sin \theta \]

2. **Substitute Known Values**:
We know that:
\[ \cos 90^\circ = 0 \quad \text{and} \quad \sin 90^\circ = 1 \]
Substituting these values into the formula gives:
\[ \cos(90^\circ + \theta) = 0 \cdot \cos \theta - 1 \cdot \sin \theta \]
This simplifies to:
\[ \cos(90^\circ + \theta) = -\sin \theta \]

3. **Find \( \sin \theta \)**:
We can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta = \frac{4}{5} \):
\[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
This simplifies to:
\[ \sin^2 \theta + \frac{16}{25} = 1 \]
Rearranging gives:
\[ \sin^2 \theta = 1 - \frac{16}{25} \]
Finding a common denominator:
\[ \sin^2 \theta = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

4. **Calculate \( \sin \theta \)**:
Taking the square root:
\[ \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]
Since \( \theta \) is in the first quadrant, \( \sin \theta \) is positive.

5. **Substitute Back to Find \( \cos(90^\circ + \theta) \)**:
Now substituting \( \sin \theta \) back into the equation for \( \cos(90^\circ + \theta) \):
\[ \cos(90^\circ + \theta) = -\sin \theta = -\frac{3}{5} \]

### Final Answer:
\[ \cos(90^\circ + \theta) = -\frac{3}{5} \]