Q.If $\cos 2\theta = 0$, then $\begin{vmatrix} 0 & \cos\theta & \sin\theta \\ \cos\theta & \sin\theta & 0 \\ \sin\theta & 0 & \cos\theta \end{vmatrix}^2 =$ ________ .

Let $a=\cos\theta,\ b=\sin\theta$. Expanding along the first row,

The condition $\cos 2\theta = 0$ gives $\cos^2\theta = \sin^2\theta = \tfrac12$. Taking the standard value $\theta=\tfrac{\pi}{4}$ (so $\cos\theta=\sin\theta=\tfrac{1}{\sqrt2}$): …

Set up

Write $a=\cos\theta$ and $b=\sin\theta$, so the determinant is

Expand along the first row

So $\Delta = -(\cos^3\theta+\sin^3\theta)$ and $\Delta^2 = (\cos^3\theta+\sin^3\theta)^2$.

Use the condition

$\cos 2\theta = 0 \Rightarrow \cos^2\theta = \sin^2\theta = \tfrac12$, hence $\cos\theta\sin\theta = \pm\tfrac12$. Expanding the square,

$\Delta^2 = \cos^6\theta+\sin^6\theta + 2\cos^3\theta\sin^3\theta = \big(1 - 3\cos^2\theta\sin^2\theta\big) + 2\cos^3\theta\sin^3\theta.$ …

Method: Evaluating a Trigonometric Determinant Under a Given Angle Condition

When a determinant's entries are trig functions of a single angle and you're given a condition on that angle (like $\cos2\theta=0$), expand the determinant symbolically first, simplify using trig identities, and only substitute the angle condition at the very end — and always check whether the condition admits more than one essentially different case.

Steps

Step 1: Expand the determinant symbolically, keeping $\cos\theta$ and $\sin\theta$ as separate variables

Use cofactor expansion along the row or column with the most zeros. Track the sign of each cofactor carefully — with several zero entries in a $3\times3$ trig determinant, it's easy to drop a minus sign on one of the surviving terms.

Step 2: Simplify the resulting expression using standard identities

The expansion typically reduces to a sum/difference of $\sin^3\theta$ and $\cos^3\theta$ (or similar). Keep the expression in terms of $\sin\theta,\cos\theta$ rather than immediately substituting numbers — this makes it easier to apply the given condition cleanly in the next step.

Step 3: Translate the given trig condition into a usable algebraic fact …

Common Mistakes

Mistake 1: Dropping a sign while cofactor-expanding a determinant with several zero entries

Why it's wrong: with three zeros scattered through a $3\times3$ trig determinant, it's tempting to write down only the surviving (non-zero) terms without also tracking each one's cofactor sign correctly — a single dropped minus sign flips the whole final expression's sign. Correct approach: write the full cofactor expansion with explicit $(-1)^{i+j}$ signs before cancelling any zero terms, not after.

Mistake 2: Assuming $\cos2\theta=0$ forces one specific value of $\cos\theta$ and $\sin\theta$ …