24 Preve that $$ 1+\frac{\cot 0}{1+\operatorname{cosec} 0}=\operatorname{cosec} heta $$

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24 Preve that $$ 1+\frac{\cot 0}{1+\operatorname{cosec} 0}=\operatorname{cosec} 	heta $$

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To prove the equation

$$
1 + \frac{\cot 0}{1 + \operatorname{cosec} 0} = \operatorname{cosec} \theta,
$$

we first need to evaluate the left-hand side of the equation.

1. **Evaluate $$\cot 0$$**:
   $$
   \cot 0 = \frac{\cos 0}{\sin 0} = \frac{1}{0},
   $$
   which is undefined.

2. **Evaluate $$\operatorname{cosec} 0$$**:
   $$
   \operatorname{cosec} 0 = \frac{1}{\sin 0} = \frac{1}{0},
   $$
   which is also undefined.

Since both $$\cot 0$$ and $$\operatorname{cosec} 0$$ are undefined, we cannot directly evaluate the left-hand side of the equation.

Thus, we need to reconsider the context of the problem. The equation might be intended to hold for some other angle $$\theta$$ rather than specifically at $$0$$.

3. **Assuming $$\theta$$ is not $$0$$**:
   If we assume $$\theta$$ is a different angle, we can rewrite the left-hand side in terms of $$\theta$$:

$$
   1 + \frac{\cot \theta}{1 + \operatorname{cosec} \theta}.
   $$

4. **Using trigonometric identities**:
   Recall that:
   $$
   \cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \operatorname{cosec} \theta = \frac{1}{\sin \theta}.
   $$

Substitute these into the left-hand side:
   $$
   1 + \frac{\frac{\cos \theta}{\sin \theta}}{1 + \frac{1}{\sin \theta}}.
   $$

5. **Simplifying the expression**:
   The denominator can be simplified:
   $$
   1 + \frac{1}{\sin \theta} = \frac{\sin \theta + 1}{\sin \theta}.
   $$

Thus, the left-hand side becomes:
   $$
   1 + \frac{\cos \theta}{\sin \theta} \cdot \frac{\sin \theta}{\sin \theta + 1} = 1 + \frac{\cos \theta}{\sin \theta + 1}.
   $$

6. **Combining terms**:
   To combine the terms, we can express $$1$$ as $$\frac{\sin \theta + 1}{\sin \theta + 1}$$:
   $$
   \frac{\sin \theta + 1}{\sin \theta + 1} + \frac{\cos \theta}{\sin \theta + 1} = \frac{\sin \theta + 1 + \cos \theta}{\sin \theta + 1}.
   $$

7. **Final expression**:
   The left-hand side simplifies to:
   $$
   \frac{\sin \theta + \cos \theta + 1}{\sin \theta + 1}.
   $$

8. **Conclusion**:
   The original equation cannot be proven as stated because it involves undefined terms at $$\theta = 0$$. However, if we consider $$\theta$$ to be any angle where $$\cot \theta$$ and $$\operatorname{cosec} \theta$$ are defined, we can derive a relationship, but it does not simplify to $$\operatorname{cosec} \theta$$ directly.

Thus, the statement as given is not valid for $$\theta = 0$$ and requires clarification or correction for other angles.
The equation $$1 + \frac{\cot 0}{1 + \operatorname{cosec} 0} = \operatorname{cosec} \theta$$ cannot be proven because $$\cot 0$$ and $$\operatorname{cosec} 0$$ are undefined.

24 Preve that

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