If `0 lt A lt pi//2` and sinA + cosA + tan A + cotA + secA + cosec A=7 and sinA and cosA are roots of equation `4x^(2) - 3x +a=0`. Then value of 25a is:

Answer

To solve the problem step by step, we start with the given equation and the conditions provided.

### Step 1: Write down the given equation
We have the equation:
\[ \sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7 \]

### Step 2: Use the properties of roots
We know that \(\sin A\) and \(\cos A\) are the roots of the quadratic equation:
\[ 4x^2 - 3x + a = 0 \]
From Vieta's formulas, we can derive two important relationships:
1. The sum of the roots \(\sin A + \cos A = -\frac{-3}{4} = \frac{3}{4}\)
2. The product of the roots \(\sin A \cos A = \frac{a}{4}\)

### Step 3: Express \(\tan A\), \(\cot A\), \(\sec A\), and \(\csc A\)
Using the definitions of the trigonometric functions:
- \(\tan A = \frac{\sin A}{\cos A}\)
- \(\cot A = \frac{\cos A}{\sin A}\)
- \(\sec A = \frac{1}{\cos A}\)
- \(\csc A = \frac{1}{\sin A}\)

### Step 4: Substitute into the equation
Now we can substitute these into the original equation:
\[ \sin A + \cos A + \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} + \frac{1}{\cos A} + \frac{1}{\sin A} = 7 \]
Let \(S = \sin A + \cos A = \frac{3}{4}\).

### Step 5: Simplify the equation
Now we can rewrite the equation:
\[ S + \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} + \frac{1}{\sin A} + \frac{1}{\cos A} = 7 \]
Since \(\sin^2 A + \cos^2 A = 1\), we have:
\[ \frac{1}{\sin A \cos A} + \frac{1}{\sin A} + \frac{1}{\cos A} = 7 - S \]
Substituting \(S = \frac{3}{4}\):
\[ \frac{1}{\sin A \cos A} + \frac{1}{\sin A} + \frac{1}{\cos A} = 7 - \frac{3}{4} = \frac{28 - 3}{4} = \frac{25}{4} \]

### Step 6: Substitute \(\sin A \cos A\)
Let \(P = \sin A \cos A = \frac{a}{4}\):
\[ \frac{4}{a} + \frac{1}{\sin A} + \frac{1}{\cos A} = \frac{25}{4} \]
Using the identities:
\[ \frac{1}{\sin A} + \frac{1}{\cos A} = \frac{\sin A + \cos A}{\sin A \cos A} = \frac{S}{P} = \frac{\frac{3}{4}}{\frac{a}{4}} = \frac{3}{a} \]
Thus, we have:
\[ \frac{4}{a} + \frac{3}{a} = \frac{25}{4} \]
Combining terms gives:
\[ \frac{7}{a} = \frac{25}{4} \]

### Step 7: Solve for \(a\)
Cross-multiplying gives:
\[ 7 \cdot 4 = 25a \implies 28 = 25a \implies a = \frac{28}{25} \]

### Step 8: Find \(25a\)
Finally, we need to find \(25a\):
\[ 25a = 25 \cdot \frac{28}{25} = 28 \]

Thus, the value of \(25a\) is:
\[ \boxed{28} \]