+134 \(Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate \[\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}.\]\)
We can use the Laplace expansion to evaluate the determinant of the given matrix. Expanding along the third column, we get:
| [[sin^2 A, cot A], [sin^2 B, cot B]] sin^2 C - [[sin^2 A, cot A], [sin^2 C, cot C]] sin^2 B + [[sin^2 B, cot B], [sin^2 C, cot C]] sin^2 A
where |M| denotes the determinant of matrix M. Note that the first term is the determinant of the 2x2 matrix [[sin^2 A, cot A], [sin^2 B, cot B]], which we can evaluate using the formula for the determinant of a 2x2 matrix:
[[sin^2 A, cot A], [sin^2 B, cot B]] = sin^2 A cot B - sin^2 B cot A
For the second and third terms, we can use the formula for the determinant of a 3x3 matrix:
[[a, b, c], [d, e, f], [g, h, i]] = a(ei - fh) - b(di - fg) + c(dh - eg)
Substituting the appropriate values, we get:
[[sin^2 A, cot A], [sin^2 C, cot C]] = sin^2 A cot C - sin^2 C cot A
[[sin^2 B, cot B], [sin^2 C, cot C]] = sin^2 B cot C - sin^2 C cot B
Substituting these values into our Laplace expansion, we get:
| [[sin^2 A, cot A], [sin^2 B, cot B], [sin^2 C, cot C]] = (sin^2 A cot B - sin^2 B cot A) sin^2 C
(sin^2 A cot C - sin^2 C cot A) sin^2 B
(sin^2 B cot C - sin^2 C cot B) sin^2 A
Factoring out sin^2 A, sin^2 B, and sin^2 C, respectively, we get:
| [[sin^2 A, cot A], [sin^2 B, cot B], [sin^2 C, cot C]] = sin^2 A (cot B sin^2 C - cot C sin^2 B)
sin^2 B (cot A sin^2 C - cot C sin^2 A)
sin^2 C (cot A sin^2 B - cot B sin^2 A)
Using the identity cot x = cos x / sin x, we can simplify the coefficients of cot A, cot B, and cot C:
cot B sin^2 C - cot C sin^2 B = cos B sin C - cos C sin B = sin(B - C) cot A sin^2 C - cot C sin^2 A = cos A sin C - cos C sin A = sin(A - C) cot A sin^2 B - cot B sin^2 A = cos A sin B - cos B sin A = sin(A - B)
Substituting these values, we get:
| [[sin^2 A, cot A], [sin^2 B, cot B], [sin^2 C, cot C]] = sin^2 A sin(B - C) - sin^2 B sin(A - C) + sin^2 C sin(A - B = 1.
Therefore, the determinant simplifies to 1.
