Question

$f\left(x\right)$= $\left|\begin{array}{ccc}{\cos }^x& \cos x\sin x& -\sin x\\ \cos x\sin x& {\sin }^x& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$
Prove that f(x)= 1 is an identity.

Answer 1

Apply $C_1-\sin x{C}_3$ and $C_2+\cos x{C}_3$

$\therefore f\left(x\right)=$ $\left|\begin{array}{ccc}1& 0& \sin x\\ 0& 1& \cos x\\ \sin x& -\cos x& 0\end{array}\right|$

Now expand and put ${\sin }^{2}x+{\cos }^{2}x=1$ for all x.

Thus f(x) =1 is an identity.