What is `((sec18^(@))/(sec144^(@))+("cosec"18^(@))/("cosec"144^(@)))` equal to ?

Options

A. `sec18^(@)`
B. `"cosec"18^(@)`
C. `-sec18^(@)`
D. `-"cosec "18^(@)`

Explanation

To solve the expression \(\frac{\sec 18^\circ}{\sec 144^\circ} + \frac{\csc 18^\circ}{\csc 144^\circ}\), we will simplify each term step by step.

### Step 1: Rewrite the secant and cosecant terms
We know that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x} \]
Thus, we can rewrite the expression as:
\[ \frac{1/\cos 18^\circ}{1/\cos 144^\circ} + \frac{1/\sin 18^\circ}{1/\sin 144^\circ} = \frac{\cos 144^\circ}{\cos 18^\circ} + \frac{\sin 144^\circ}{\sin 18^\circ} \]

### Step 2: Use angle identities
We know that:
\[ \cos(180^\circ - x) = -\cos x \quad \text{and} \quad \sin(180^\circ - x) = \sin x \]
Thus:
\[ \cos 144^\circ = \cos(180^\circ - 36^\circ) = -\cos 36^\circ \]
\[ \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ \]
Substituting these into the expression gives:
\[ \frac{-\cos 36^\circ}{\cos 18^\circ} + \frac{\sin 36^\circ}{\sin 18^\circ} \]

### Step 3: Combine the fractions
Now we can combine the two fractions:
\[ -\frac{\cos 36^\circ}{\cos 18^\circ} + \frac{\sin 36^\circ}{\sin 18^\circ} = \frac{-\cos 36^\circ \sin 18^\circ + \sin 36^\circ \cos 18^\circ}{\cos 18^\circ \sin 18^\circ} \]

### Step 4: Use the sine difference identity
We can use the sine difference identity:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
Thus:
\[ -\cos 36^\circ \sin 18^\circ + \sin 36^\circ \cos 18^\circ = \sin(36^\circ - 18^\circ) = \sin 18^\circ \]
So, we have:
\[ \frac{\sin 18^\circ}{\cos 18^\circ \sin 18^\circ} = \frac{1}{\cos 18^\circ} = \sec 18^\circ \]

### Final Result
Thus, the final result is:
\[ \frac{\sec 18^\circ}{\sec 144^\circ} + \frac{\csc 18^\circ}{\csc 144^\circ} = \sec 18^\circ \]