Find `min^(m)` value of
`sin^(2)alpha +cosec^(2)alpha+cos^(2)beta+sec^(2)beta+tan^(2)gamma+cot^(2)gamma=?`

Answer

To find the minimum value of the expression \( \sin^2 \alpha + \csc^2 \alpha + \cos^2 \beta + \sec^2 \beta + \tan^2 \gamma + \cot^2 \gamma \), we will use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Identify the Terms**:
The expression consists of six terms:
\[ \sin^2 \alpha, \quad \csc^2 \alpha, \quad \cos^2 \beta, \quad \sec^2 \beta, \quad \tan^2 \gamma, \quad \cot^2 \gamma \]

2. **Apply AM-GM Inequality**:
According to the AM-GM inequality, for any non-negative real numbers \( a_1, a_2, \ldots, a_n \):
\[ \frac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdot \ldots \cdot a_n} \]
Here, we have six terms, so:
\[ \frac{\sin^2 \alpha + \csc^2 \alpha + \cos^2 \beta + \sec^2 \beta + \tan^2 \gamma + \cot^2 \gamma}{6} \geq \sqrt[6]{\sin^2 \alpha \cdot \csc^2 \alpha \cdot \cos^2 \beta \cdot \sec^2 \beta \cdot \tan^2 \gamma \cdot \cot^2 \gamma} \]

3. **Calculate the Product**:
We know:
\[ \csc^2 \alpha = \frac{1}{\sin^2 \alpha}, \quad \sec^2 \beta = \frac{1}{\cos^2 \beta}, \quad \tan^2 \gamma = \frac{\sin^2 \gamma}{\cos^2 \gamma}, \quad \cot^2 \gamma = \frac{\cos^2 \gamma}{\sin^2 \gamma} \]
Therefore, the product becomes:
\[ \sin^2 \alpha \cdot \csc^2 \alpha = 1, \quad \cos^2 \beta \cdot \sec^2 \beta = 1, \quad \tan^2 \gamma \cdot \cot^2 \gamma = 1 \]
Thus:
\[ \sin^2 \alpha \cdot \csc^2 \alpha \cdot \cos^2 \beta \cdot \sec^2 \beta \cdot \tan^2 \gamma \cdot \cot^2 \gamma = 1 \cdot 1 \cdot 1 = 1 \]

4. **Substitute Back into AM-GM**:
Now substituting back into the AM-GM inequality:
\[ \frac{\sin^2 \alpha + \csc^2 \alpha + \cos^2 \beta + \sec^2 \beta + \tan^2 \gamma + \cot^2 \gamma}{6} \geq \sqrt[6]{1} = 1 \]
Therefore:
\[ \sin^2 \alpha + \csc^2 \alpha + \cos^2 \beta + \sec^2 \beta + \tan^2 \gamma + \cot^2 \gamma \geq 6 \]

5. **Conclusion**:
The minimum value of the expression \( \sin^2 \alpha + \csc^2 \alpha + \cos^2 \beta + \sec^2 \beta + \tan^2 \gamma + \cot^2 \gamma \) is \( 6 \).