What is the value of cos 2 67∘ sin 2 23∘ ?

Q. Without expanding, show that the values of each of the following determinants are zero:
(i)

|\begin{matrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{matrix}|

(ii)

|\begin{matrix} 6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2 \end{matrix}|

(iii)

|\begin{matrix} 2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12 \end{matrix}|

(iv)

|\begin{matrix} 1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b \end{matrix}|

(v)

|\begin{matrix} a + b & 2 a + b & 3 a + b \\ 2 a + b & 3 a + b & 4 a + b \\ 4 a + b & 5 a + b & 6 a + b \end{matrix}|

(vi)

|\begin{matrix} 1 & a & a^{2} - b c \\ 1 & b & b^{2} - a c \\ 1 & c & c^{2} - a b \end{matrix}|

(vii)

|\begin{matrix} 49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3 \end{matrix}|

(viii)

|\begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix}|

(ix)

|\begin{matrix} 1 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2 \end{matrix}|

(x)

|\begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} \\ 2^{2} & 3^{2} & 4^{2} & 5^{2} \\ 3^{2} & 4^{2} & 5^{2} & 6^{2} \\ 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix}|

(xi)

|\begin{matrix} a & b & c \\ a + 2 x & b + 2 y & c + 2 z \\ x & y & z \end{matrix}|

(xii)

|\begin{matrix} {(2^{x} + 2^{- x})}^{2} & {(2^{x} - 2^{- x})}^{2} & 1 \\ {(3^{x} + 3^{- x})}^{2} & {(3^{x} - 3^{- x})}^{2} & 1 \\ {(4^{x} + 4^{- x})}^{2} & {(4^{x} - 4^{- x})}^{2} & 1 \end{matrix}|

(xiii)

|\begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix}|

(xiv)

|\begin{matrix} \sin^{2} 23 ° & \sin^{2} 67 ° & \cos 180 ° \\ - \sin^{2} 67 ° & - \sin^{2} 23 ° & \cos^{2} 180 ° \\ \cos 180 ° & \sin^{2} 23 ° & \sin^{2} 67 ° \end{matrix}|

(xv)

|\begin{matrix} \cos (x + y) & - \sin (x + y) & \cos 2 y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y \end{matrix}|

(xvi)

|\begin{matrix} \sqrt{23} + \sqrt{3} & \sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{46} & 5 & \sqrt{10} \\ 3 + \sqrt{115} & \sqrt{15} & 5 \end{matrix}|

(xvii)

|\begin{matrix} \sin^{2} A & \cot A & 1 \\ \sin^{2} B & \cot B & 1 \\ \sin^{2} C & \cot C & 1 \end{matrix}|, where A, B, C are the angles of ∆ A B C .

What is the value of cos² 67° – sin² 23°?