The value of `cosec 430 ^(@) + sqrt3 sec 470^(@)` is :

Options

A. 1
B. 1
C. `-4`
D. 4

Answer

To solve the expression \( \csc 430^\circ + \sqrt{3} \sec 470^\circ \), we will follow these steps:

### Step 1: Rewrite the expression in terms of sine and cosine
The cosecant and secant functions can be rewritten as:
\[ \csc 430^\circ = \frac{1}{\sin 430^\circ} \quad \text{and} \quad \sec 470^\circ = \frac{1}{\cos 470^\circ} \]
Thus, the expression becomes:
\[ \frac{1}{\sin 430^\circ} + \sqrt{3} \cdot \frac{1}{\cos 470^\circ} \]

### Step 2: Simplify the angles
Since the angles are greater than 360 degrees, we can reduce them:
\[ 430^\circ - 360^\circ = 70^\circ \quad \text{and} \quad 470^\circ - 360^\circ = 110^\circ \]
So, we can rewrite the expression as:
\[ \frac{1}{\sin 70^\circ} + \sqrt{3} \cdot \frac{1}{\cos 110^\circ} \]

### Step 3: Use the identity for cosine
We know that:
\[ \cos 110^\circ = \cos(180^\circ - 70^\circ) = -\cos 70^\circ \]
Thus, we can rewrite the expression:
\[ \frac{1}{\sin 70^\circ} - \sqrt{3} \cdot \frac{1}{\cos 70^\circ} \]

### Step 4: Combine the fractions
Now we can combine the two terms over a common denominator:
\[ \frac{1 \cdot \cos 70^\circ - \sqrt{3} \cdot \sin 70^\circ}{\sin 70^\circ \cos 70^\circ} \]

### Step 5: Rewrite the denominator
Using the double angle identity, we know:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Thus, we can write:
\[ \sin 70^\circ \cos 70^\circ = \frac{1}{2} \sin 140^\circ \]
So our expression becomes:
\[ \frac{\cos 70^\circ - \sqrt{3} \sin 70^\circ}{\frac{1}{2} \sin 140^\circ} = \frac{2(\cos 70^\circ - \sqrt{3} \sin 70^\circ)}{\sin 140^\circ} \]

### Step 6: Simplify the numerator
We can express the numerator in terms of sine:
\[ \cos 70^\circ - \sqrt{3} \sin 70^\circ = 2 \left( \frac{1}{2} \cos 70^\circ - \frac{\sqrt{3}}{2} \sin 70^\circ \right) \]
This can be recognized as:
\[ 2 \sin\left(70^\circ - 30^\circ\right) = 2 \sin 40^\circ \]
Thus, the expression simplifies to:
\[ \frac{2 \cdot 2 \sin 40^\circ}{\sin 140^\circ} \]

### Step 7: Use the identity for sine
Since \( \sin 140^\circ = \sin(180^\circ - 40^\circ) = \sin 40^\circ \):
\[ \frac{4 \sin 40^\circ}{\sin 40^\circ} = 4 \]

### Final Result
Thus, the value of the expression \( \csc 430^\circ + \sqrt{3} \sec 470^\circ \) is:
\[ \boxed{4} \]