Calculate the value of A, if :
`(sec2A-1)(cosec3A-1)=0`

Answer

To solve the equation \((\sec 2A - 1)(\csc 3A - 1) = 0\), we will break it down into two separate equations and solve for \(A\).

### Step 1: Set each factor to zero
We start with the equation:
\[ (\sec 2A - 1)(\csc 3A - 1) = 0 \]
This implies that either:
\[ \sec 2A - 1 = 0 \quad \text{or} \quad \csc 3A - 1 = 0 \]

### Step 2: Solve \(\sec 2A - 1 = 0\)
From the first equation:
\[ \sec 2A = 1 \]
We know that \(\sec \theta = 1\) when \(\theta = 0^\circ + n \cdot 360^\circ\) (where \(n\) is any integer). Thus, we have:
\[ 2A = 0^\circ + n \cdot 360^\circ \]
Dividing by 2:
\[ A = 0^\circ + \frac{n \cdot 360^\circ}{2} = 0^\circ + n \cdot 180^\circ \]
So, one solution is:
\[ A = 0^\circ \]

### Step 3: Solve \(\csc 3A - 1 = 0\)
From the second equation:
\[ \csc 3A = 1 \]
We know that \(\csc \theta = 1\) when \(\theta = 90^\circ + m \cdot 360^\circ\) (where \(m\) is any integer). Thus, we have:
\[ 3A = 90^\circ + m \cdot 360^\circ \]
Dividing by 3:
\[ A = 30^\circ + m \cdot 120^\circ \]
So, another solution is:
\[ A = 30^\circ \]

### Step 4: Combine the solutions
From both equations, we have the solutions:
\[ A = 0^\circ \quad \text{and} \quad A = 30^\circ \]

### Final Answer
The values of \(A\) are:
\[ A = 0^\circ \quad \text{and} \quad A = 30^\circ \]

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