Evaluate `(sec A + tan A) (1- sin A)` for `A = 60^(@)`.

Answer

To evaluate the expression \((\sec A + \tan A)(1 - \sin A)\) for \(A = 60^\circ\), we will follow these steps:

### Step 1: Substitute \(A = 60^\circ\) into the expression
We start with the expression:
\[ (\sec 60^\circ + \tan 60^\circ)(1 - \sin 60^\circ) \]

### Step 2: Calculate \(\sec 60^\circ\), \(\tan 60^\circ\), and \(\sin 60^\circ\)
Using trigonometric values:
- \(\sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2\)
- \(\tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)

### Step 3: Substitute the values into the expression
Now we substitute these values back into the expression:
\[ (2 + \sqrt{3})(1 - \frac{\sqrt{3}}{2}) \]

### Step 4: Simplify \(1 - \sin 60^\circ\)
Calculating \(1 - \sin 60^\circ\):
\[ 1 - \frac{\sqrt{3}}{2} = \frac{2}{2} - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \]

### Step 5: Multiply the two parts
Now we multiply:
\[ (2 + \sqrt{3}) \left(\frac{2 - \sqrt{3}}{2}\right) \]
This can be simplified as:
\[ \frac{(2 + \sqrt{3})(2 - \sqrt{3})}{2} \]

### Step 6: Use the difference of squares
Using the difference of squares formula \(a^2 - b^2\):
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]
Thus, we have:
\[ \frac{1}{2} \]

### Final Answer
The value of the expression \((\sec 60^\circ + \tan 60^\circ)(1 - \sin 60^\circ)\) is:
\[ \frac{1}{2} \]