Value of the expression : `(1+2sin60^(@)cos60^(@))/(sin60^(@)+cos60^(@))+(1-2sin60^(@)cos60^(@))/(sin60^(@)-cos60^(@))` is

Options

A. `2sqrt(3)`
B. 0
C. `sqrt(3)`
D. 2

Answer

To solve the expression

\[ \frac{1 + 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ + \cos 60^\circ} + \frac{1 - 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ - \cos 60^\circ} \]

we will break it down into two parts, \( P \) and \( Q \).

### Step 1: Calculate \( P \)

Let

\[ P = \frac{1 + 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ + \cos 60^\circ} \]

Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can rewrite \( 1 \) as \( \sin^2 60^\circ + \cos^2 60^\circ \):

\[ P = \frac{\sin^2 60^\circ + \cos^2 60^\circ + 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ + \cos 60^\circ} \]

Recognizing \( \sin^2 60^\circ + \cos^2 60^\circ + 2 \sin 60^\circ \cos 60^\circ \) as \( (\sin 60^\circ + \cos 60^\circ)^2 \):

\[ P = \frac{(\sin 60^\circ + \cos 60^\circ)^2}{\sin 60^\circ + \cos 60^\circ} \]

This simplifies to:

\[ P = \sin 60^\circ + \cos 60^\circ \]

### Step 2: Calculate \( \sin 60^\circ \) and \( \cos 60^\circ \)

Using known values:

\[ \sin 60^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2} \]

Thus,

\[ P = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2} \]

### Step 3: Calculate \( Q \)

Let

\[ Q = \frac{1 - 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ - \cos 60^\circ} \]

Similarly, we can rewrite \( 1 \) as \( \sin^2 60^\circ + \cos^2 60^\circ \):

\[ Q = \frac{\sin^2 60^\circ + \cos^2 60^\circ - 2 \sin 60^\circ \cos 60^\circ}{\sin 60^\circ - \cos 60^\circ} \]

Recognizing \( \sin^2 60^\circ + \cos^2 60^\circ - 2 \sin 60^\circ \cos 60^\circ \) as \( (\sin 60^\circ - \cos 60^\circ)^2 \):

\[ Q = \frac{(\sin 60^\circ - \cos 60^\circ)^2}{\sin 60^\circ - \cos 60^\circ} \]

This simplifies to:

\[ Q = \sin 60^\circ - \cos 60^\circ \]

### Step 4: Calculate \( Q \)

Using the values again:

\[ Q = \frac{\sqrt{3}}{2} - \frac{1}{2} = \frac{\sqrt{3} - 1}{2} \]

### Step 5: Combine \( P \) and \( Q \)

Now, we add \( P \) and \( Q \):

\[ P + Q = \left( \frac{\sqrt{3} + 1}{2} \right) + \left( \frac{\sqrt{3} - 1}{2} \right) \]

Combining the fractions:

\[ P + Q = \frac{(\sqrt{3} + 1) + (\sqrt{3} - 1)}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Final Answer

Thus, the value of the expression is

\[ \sqrt{3} \]