Q1(iii):

Evaluate the following : (iii) $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$

Given: The trigonometric expression $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$.

To Find: The numerical value of the given expression.

Visual Representation (Trigonometric Ratios):

Step 1: Identify the values of the trigonometric ratios.
Using the standard trigonometric table for specific angles:

• $\cos 45^\circ = \frac{1}{\sqrt{2}}$
• $\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$
• $\text{cosec } 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{1/2} = 2$

Step 2: Substitute the values into the expression.
The expression is $\frac{\cos 45^\circ}{\sec 30^\circ + \text{cosec } 30^\circ}$. Substituting the values:

$\text{Expression} = \frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}} + 2}$

Step 3: Simplify the denominator.
To add the terms in the denominator, find a common denominator:

$\frac{2}{\sqrt{3}} + 2 = \frac{2 + 2\sqrt{3}}{\sqrt{3}} = \frac{2(1 + \sqrt{3})}{\sqrt{3}}$

Step 4: Perform the division of fractions.
$\text{Expression} = \frac{1}{\sqrt{2}} \div \frac{2(1 + \sqrt{3})}{\sqrt{3}}$
$\text{Expression} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2(1 + \sqrt{3})}$
$\text{Expression} = \frac{\sqrt{3}}{2\sqrt{2}(1 + \sqrt{3})} = \frac{\sqrt{3}}{2(\sqrt{2} + \sqrt{6})}$

Step 5: Rationalize the denominator.
Multiply the numerator and denominator by the conjugate $(\sqrt{6} - \sqrt{2})$:

$\text{Expression} = \frac{\sqrt{3}}{2(\sqrt{6} + \sqrt{2})} \times \frac{\sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}$
$= \frac{\sqrt{18} - \sqrt{6}}{2((\sqrt{6})^2 - (\sqrt{2})^2)}$
$= \frac{3\sqrt{2} - \sqrt{6}}{2(6 - 2)}$
$= \frac{3\sqrt{2} - \sqrt{6}}{2(4)}$
$= \frac{3\sqrt{2} - \sqrt{6}}{8}$

Final Answer: $\frac{3\sqrt{2} - \sqrt{6}}{8}$