24 Preve that \( 1+\frac{\cot 0}{1+\operatorname{cosec} 0}=\operatorname{cosec} \theta \)

To prove that \[ 1 + \frac{\cot 0}{1 + \operatorname{cosec} 0} = \operatorname{cosec} \theta, \] let's analyze the terms on the left-hand side. First, we note that \(\cot 0 = \frac{\cos 0}{\sin 0} = \frac{1}{0}\), which is undefined. So, let's keep this in mind when evaluating. Next, for \(\operatorname{cosec} 0\), we have: \[ \operatorname{cosec} 0 = \frac{1}{\sin 0} = \frac{1}{0}, \] which is also undefined. Given both these values are undefined, the expression on the left side cannot be evaluated in a standard way. Therefore, the equation does not hold true as both sides become undefined when evaluated under regular definitions. If this is meant to validate relationships in the context of a general angle \(\theta\) where \(\theta \neq 0\), we might need to redefine or reconsider the problem context. Thus, we cannot validate this expression \(1+\frac{\cot 0}{1+\operatorname{cosec} 0}=\operatorname{cosec} \theta\) because both terms on the left are undefined at \(0\). Hence, this equation does not hold.