Question

Find the the value given determinant $\left| {\begin{array}{*{20}{c}} {\cos {{15}^ \circ }}&{\sin {{15}^ \circ }} \\ {\sin {{75}^ \circ }}&{\cos {{75}^ \circ }} \end{array}} \right|$ A.$1$ B. $0$ C. $2$ D. $3$

Hint: This question can be solved by simply solving the determinant.

Given determinant is
$\left| {\begin{array}{*{20}{c}} {\cos {{15}^ \circ }}&{\sin {{15}^ \circ }} \\ {\sin {{75}^ \circ }}&{\cos {{75}^ \circ }} \end{array}} \right|$
Now on solving the determinant we get,
$\cos {75^ \circ } \cdot \cos {15^ \circ } - \sin {75^ \circ } \cdot \sin {15^ \circ }$
Now we know that,
$\cos \left( {A + B} \right) = \cos A \cdot \cos B - \sin A \cdot \sin B$
Using the above equation we get,
$\cos \left( {{{75}^ \circ } + {{15}^ \circ }} \right) \\ {\text{or }}\cos \left( {{{90}^ \circ }} \right) \\ = 0 \\$
Therefore, the correct option is (B).

Note: These types of questions can be solved by simply solving the determinant. Here in this question we simply solve the determinant and then we apply the formula of $\cos \left( {A + B} \right)$ and then we get our answer.