Compute the square of the value of the expression `(4+sec 20^@)/(cosec 20^@)`

Answer

To compute the square of the value of the expression \((4 + \sec 20^\circ) / \csc 20^\circ\), we can follow these steps:

### Step 1: Define the Expression
Let \( y = \frac{4 + \sec 20^\circ}{\csc 20^\circ} \).

### Step 2: Cross Multiply
Cross multiplying gives us:
\[ y \cdot \csc 20^\circ = 4 + \sec 20^\circ \]

### Step 3: Rewrite in Terms of Sine and Cosine
Recall that \(\csc 20^\circ = \frac{1}{\sin 20^\circ}\) and \(\sec 20^\circ = \frac{1}{\cos 20^\circ}\). Substituting these into the equation gives:
\[ y \cdot \frac{1}{\sin 20^\circ} = 4 + \frac{1}{\cos 20^\circ} \]

### Step 4: Multiply Through by \(\sin 20^\circ \cos 20^\circ\)
Multiply both sides by \(\sin 20^\circ \cos 20^\circ\):
\[ y \cos 20^\circ = 4 \sin 20^\circ + \sin 20^\circ \]

### Step 5: Simplify the Right Side
This simplifies to:
\[ y \cos 20^\circ = 4 \sin 20^\circ + \sin 20^\circ = 5 \sin 20^\circ \]

### Step 6: Solve for \(y\)
Now, divide both sides by \(\cos 20^\circ\):
\[ y = \frac{5 \sin 20^\circ}{\cos 20^\circ} = 5 \tan 20^\circ \]

### Step 7: Find the Square of \(y\)
Now we need to find \(y^2\):
\[ y^2 = (5 \tan 20^\circ)^2 = 25 \tan^2 20^\circ \]

### Step 8: Final Result
Thus, the square of the value of the expression is:
\[ y^2 = 25 \tan^2 20^\circ \]