`|["cos"70^(@), "sin"20^(@)], ["sin"70^(@), "cos"20^(@)]|=?`

Options

A. `1`
B. `0`
C. `"cos"50^(@)`
D. `"sin"50^(@)`

Answer

To solve the determinant

\[ D = \begin{vmatrix} \cos 70^\circ & \sin 20^\circ \\ \sin 70^\circ & \cos 20^\circ \end{vmatrix} \]

we will follow these steps:

### Step 1: Calculate the determinant using the formula
The formula for the determinant of a 2x2 matrix

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

is given by \( ad - bc \). Here, we have:

- \( a = \cos 70^\circ \)
- \( b = \sin 20^\circ \)
- \( c = \sin 70^\circ \)
- \( d = \cos 20^\circ \)

So, we can write:

\[ D = \cos 70^\circ \cdot \cos 20^\circ - \sin 20^\circ \cdot \sin 70^\circ \]

### Step 2: Apply the cosine addition formula
We recognize that the expression we obtained resembles the cosine of a sum. The cosine addition formula states:

\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

In our case, let \( A = 70^\circ \) and \( B = 20^\circ \). Therefore, we have:

\[ D = \cos(70^\circ + 20^\circ) = \cos 90^\circ \]

### Step 3: Evaluate \(\cos 90^\circ\)
We know that:

\[ \cos 90^\circ = 0 \]

### Final Answer
Thus, the value of the determinant is:

\[ D = 0 \]

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