`sec70^(@)sin20^(@)+cos20^(@)"cosec"70^(@)=?`

Options

A. `0`
B. `1`
C. `-1`
D. `2`

Explanation

To solve the expression \( \sec 70^\circ \sin 20^\circ + \cos 20^\circ \csc 70^\circ \), we will use trigonometric identities and properties.

### Step-by-Step Solution:

1. **Rewrite secant and cosecant**:
\[ \sec 70^\circ = \frac{1}{\cos 70^\circ} \quad \text{and} \quad \csc 70^\circ = \frac{1}{\sin 70^\circ} \]
Thus, we can rewrite the expression as:
\[ \frac{\sin 20^\circ}{\cos 70^\circ} + \cos 20^\circ \cdot \frac{1}{\sin 70^\circ} \]

2. **Use the co-function identities**:
We know that:
\[ \cos 70^\circ = \sin 20^\circ \quad \text{and} \quad \sin 70^\circ = \cos 20^\circ \]
Substituting these into the expression gives:
\[ \frac{\sin 20^\circ}{\sin 20^\circ} + \cos 20^\circ \cdot \frac{1}{\cos 20^\circ} \]

3. **Simplify the fractions**:
The first term simplifies to:
\[ 1 \]
The second term simplifies to:
\[ 1 \]
Therefore, we have:
\[ 1 + 1 = 2 \]

4. **Final answer**:
Thus, the value of the expression \( \sec 70^\circ \sin 20^\circ + \cos 20^\circ \csc 70^\circ \) is:
\[ \boxed{2} \]