If `0^(@)lt theta lt 90^(@)` and `cos theta=(4)/(5)` find the values of
`"cosec"(180^(@)+theta)`

Answer

To solve the problem, we need to find the value of \( \csc(180^\circ + \theta) \) given that \( \cos \theta = \frac{4}{5} \) and \( 0^\circ < \theta < 90^\circ \).

### Step-by-step Solution:

1. **Find \( \sin \theta \)**:
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \).
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting \( \cos \theta = \frac{4}{5} \):
\[ \sin^2 \theta = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

2. **Determine \( \csc(180^\circ + \theta) \)**:
The cosecant function is the reciprocal of the sine function:
\[ \csc(180^\circ + \theta) = \frac{1}{\sin(180^\circ + \theta)} \]
We know that:
\[ \sin(180^\circ + \theta) = -\sin \theta \]
Therefore:
\[ \csc(180^\circ + \theta) = \frac{1}{-\sin \theta} = -\frac{1}{\sin \theta} \]

3. **Substituting the value of \( \sin \theta \)**:
Now substituting \( \sin \theta = \frac{3}{5} \):
\[ \csc(180^\circ + \theta) = -\frac{1}{\frac{3}{5}} = -\frac{5}{3} \]

### Final Answer:
\[ \csc(180^\circ + \theta) = -\frac{5}{3} \]