Question

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

Answer 1

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

Given determinant is
 $\left|\begin{array}{cc}\cos {15}^{\circ }& \sin {15}^{\circ }\\ \sin {75}^{\circ }& \cos {75}^{\circ }\end{array}\right|$
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,
$$\begin{gathered} \cos \left({75}^{\circ }+{15}^{\circ }\right) \\ \text{or }\cos \left({90}^{\circ }\right) \\ =0 \end{gathered}$$
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.