If `3cotA=4`, find the value of `("Cosec"^(2)A+1)/("Cosec"^(2)A-1)`.

Answer

To solve the problem, we start with the given equation:

**Step 1: Start with the given equation.**
We have:
\[ 3 \cot A = 4 \]

**Step 2: Solve for cot A.**
Dividing both sides by 3, we get:
\[ \cot A = \frac{4}{3} \]

**Step 3: Square both sides to find cot² A.**
Now, squaring both sides:
\[ \cot^2 A = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]

**Step 4: Use the trigonometric identity.**
We use the identity:
\[ \csc^2 A - \cot^2 A = 1 \]
This can be rearranged to find \(\csc^2 A\):
\[ \csc^2 A = 1 + \cot^2 A \]

**Step 5: Substitute cot² A into the identity.**
Substituting \(\cot^2 A\) into the equation:
\[ \csc^2 A = 1 + \frac{16}{9} \]
To add these, convert 1 to a fraction:
\[ 1 = \frac{9}{9} \]
Thus:
\[ \csc^2 A = \frac{9}{9} + \frac{16}{9} = \frac{25}{9} \]

**Step 6: Substitute \(\csc^2 A\) into the expression.**
Now we need to find:
\[ \frac{\csc^2 A + 1}{\csc^2 A - 1} \]
Substituting \(\csc^2 A\):
\[ \frac{\frac{25}{9} + 1}{\frac{25}{9} - 1} \]
Convert 1 to a fraction:
\[ 1 = \frac{9}{9} \]
So:
\[ \frac{\frac{25}{9} + \frac{9}{9}}{\frac{25}{9} - \frac{9}{9}} = \frac{\frac{34}{9}}{\frac{16}{9}} \]

**Step 7: Simplify the fraction.**
Now, simplifying:
\[ \frac{\frac{34}{9}}{\frac{16}{9}} = \frac{34}{16} \]
This can be simplified further:
\[ \frac{34 \div 2}{16 \div 2} = \frac{17}{8} \]

**Final Answer:**
Thus, the value of \(\frac{\csc^2 A + 1}{\csc^2 A - 1}\) is:
\[ \frac{17}{8} \]

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