Let `a , b`and `c` detnote the sides `BC,CA `and`AB` respectively of triangle` ABC`. If `|[1,a, b],[1,c, a],[1,b ,c]|=0`. find the value of `sin^2A+sin^2B+sin^2C`

Explanation

To solve the problem, we need to find the value of \( \sin^2 A + \sin^2 B + \sin^2 C \) given that the determinant \( |[1, a, b], [1, c, a], [1, b, c]| = 0 \), where \( a, b, c \) denote the sides \( BC, CA, AB \) respectively of triangle \( ABC \).

### Step 1: Understanding the Determinant Condition
The condition \( |[1, a, b], [1, c, a], [1, b, c]| = 0 \) implies that the rows of the determinant are linearly dependent. This means that the points represented by these rows are collinear.

### Step 2: Expanding the Determinant
We can expand the determinant as follows:

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