If `theta=45^(@)`, then find the value of `"cosec"^(2)theta`.

Answer

To find the value of \(\csc^2 \theta\) when \(\theta = 45^\circ\), we can follow these steps:

### Step 1: Understand the definition of cosecant
The cosecant function is defined as:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Thus, to find \(\csc^2 \theta\), we need to first find \(\sin \theta\).

### Step 2: Find \(\sin 45^\circ\)
We know from trigonometric ratios that:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 3: Calculate \(\csc 45^\circ\)
Using the definition of cosecant:
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 4: Calculate \(\csc^2 45^\circ\)
Now, we square the value of \(\csc 45^\circ\):
\[ \csc^2 45^\circ = (\sqrt{2})^2 = 2 \]

### Final Answer
Thus, the value of \(\csc^2 \theta\) when \(\theta = 45^\circ\) is:
\[ \csc^2 45^\circ = 2 \]

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