Express the following as functions of angles less than `45^(@)` :
`"cosec"(-7498^(@))`.

Answer

To express \( \csc(-7498^\circ) \) as a function of angles less than \( 45^\circ \), we can follow these steps:

### Step 1: Use the property of cosecant for negative angles
We know that:
\[ \csc(-\theta) = -\csc(\theta) \]
Thus,
\[ \csc(-7498^\circ) = -\csc(7498^\circ) \]

### Step 2: Reduce the angle \( 7498^\circ \) to an equivalent angle within \( 0^\circ \) to \( 360^\circ \)
To find an equivalent angle, we can subtract multiples of \( 360^\circ \) from \( 7498^\circ \):
\[ 7498^\circ - 360^\circ \times 20 = 7498^\circ - 7200^\circ = 298^\circ \]
So, we have:
\[ \csc(7498^\circ) = \csc(298^\circ) \]

### Step 3: Use the property of cosecant for angles greater than \( 270^\circ \)
Next, we can express \( \csc(298^\circ) \) in terms of an angle less than \( 45^\circ \):
\[ \csc(298^\circ) = \csc(270^\circ + 28^\circ) \]
Using the identity:
\[ \csc(270^\circ + \theta) = -\sec(\theta) \]
we find:
\[ \csc(298^\circ) = -\sec(28^\circ) \]

### Step 4: Combine the results
Now substituting back, we have:
\[ \csc(-7498^\circ) = -\csc(7498^\circ) = -\csc(298^\circ) = -(-\sec(28^\circ)) = \sec(28^\circ) \]

### Final Result
Thus, we conclude:
\[ \csc(-7498^\circ) = \sec(28^\circ) \]

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