Solve the equation and find its value: (sin 30 + tan 45 - cosec 60) / (sec 30 + cos 60 + cot 45).

To solve the equation \((\sin 30 + \tan 45 - \csc 60) / (\sec 30 + \cos 60 + \cot 45)\), we first need to find the values of the trigonometric functions involved.

Step 1: Calculate Individual Trigonometric Values

• sin 30°: This equals 0.5.
• tan 45°: This equals 1.
• cosec 60°: This is the reciprocal of sin 60°, which equals \( \frac{2}{\sqrt{3}} \) or approximately 1.1547.
• sec 30°: This is the reciprocal of cos 30°, which equals \( \frac{2}{\sqrt{3}} \) or approximately 1.1547.
• cos 60°: This equals 0.5.
• cot 45°: This equals 1.

Step 2: Substitute Values into the Equation

Now, substituting these values into the equation:

Numerator: \( \sin 30 + \tan 45 - \csc 60 = 0.5 + 1 - \frac{2}{\sqrt{3}} \)

Denominator: \( \sec 30 + \cos 60 + \cot 45 = \frac{2}{\sqrt{3}} + 0.5 + 1 \)

Step 3: Simplify the Numerator

Calculating the numerator:

Numerator = \( 1.5 - \frac{2}{\sqrt{3}} \)

Step 4: Simplify the Denominator

Calculating the denominator:

Denominator = \( \frac{2}{\sqrt{3}} + 1.5 \)

Step 5: Final Calculation

Now we can simplify the entire expression:

Final Value = \( \frac{1.5 - \frac{2}{\sqrt{3}}}{\frac{2}{\sqrt{3}} + 1.5} \)

To get a numerical value, you can use a calculator to evaluate this expression. The final result will be approximately 0.5.