Evaluating Trigonometric Expression with Standard Angles

The question asks us to find the value of a given trigonometric expression:

\[ \frac{\tan^2 30^\circ + \sin^2 90^\circ + \cot^2 60^\circ + \sin^2 30^\circ \cos^2 45^\circ}{\sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ} \]

To evaluate this expression, we need to know the values of the trigonometric functions for the standard angles \(30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\).

Standard Trigonometric Values

Here are the standard values we will use:

Evaluate the Numerator

The numerator is \(\tan^2 30^\circ + \sin^2 90^\circ + \cot^2 60^\circ + \sin^2 30^\circ \cos^2 45^\circ\).

Substitute the standard values:

• \(\tan 30^\circ = \frac{1}{\sqrt{3}}\), so \(\tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\).
• \(\sin 90^\circ = 1\), so \(\sin^2 90^\circ = (1)^2 = 1\).
• \(\cot 60^\circ = \frac{1}{\sqrt{3}}\), so \(\cot^2 60^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}\).
• \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 45^\circ = \frac{1}{\sqrt{2}}\), so \(\sin^2 30^\circ \cos^2 45^\circ = \left(\frac{1}{2}\right)^2 \times \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\).

Now, add these values together for the numerator:

\[ \text{Numerator} = \frac{1}{3} + 1 + \frac{1}{3} + \frac{1}{8} \]

Combine the fractions with like denominators:

\[ \text{Numerator} = \left(\frac{1}{3} + \frac{1}{3}\right) + 1 + \frac{1}{8} = \frac{2}{3} + 1 + \frac{1}{8} \]

To add these, find a common denominator, which is 24:

\[ \text{Numerator} = \frac{2 \times 8}{3 \times 8} + \frac{1 \times 24}{1 \times 24} + \frac{1 \times 3}{8 \times 3} = \frac{16}{24} + \frac{24}{24} + \frac{3}{24} \]

\[ \text{Numerator} = \frac{16 + 24 + 3}{24} = \frac{43}{24} \]

Evaluate the Denominator

The denominator is \(\sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ\).

Substitute the standard values:

• \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
• \(\cos 60^\circ = \frac{1}{2}\)
• \(\sin 30^\circ = \frac{1}{2}\)

So, the denominator is:

\[ \text{Denominator} = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) \]

\[ \text{Denominator} = \frac{\sqrt{3} \times \sqrt{3}}{2 \times 2} - \frac{1 \times 1}{2 \times 2} = \frac{3}{4} - \frac{1}{4} \]

\[ \text{Denominator} = \frac{3 - 1}{4} = \frac{2}{4} = \frac{1}{2} \]

Alternatively, we can recognise the trigonometric identity \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). The denominator has the form \(\sin 60^\circ \cos 30^\circ - \cos 60^\circ \sin 30^\circ\), which is \(\sin(60^\circ - 30^\circ) = \sin 30^\circ\). The value of \(\sin 30^\circ\) is \(\frac{1}{2}\). Both methods give the same result for the denominator.

Calculate the Final Value of the Expression

The expression is \(\frac{\text{Numerator}}{\text{Denominator}}\).

Substitute the calculated values:

\[ \text{Value} = \frac{\frac{43}{24}}{\frac{1}{2}} \]

To divide by a fraction, multiply by its reciprocal:

\[ \text{Value} = \frac{43}{24} \times \frac{2}{1} = \frac{43 \times 2}{24 \times 1} = \frac{86}{24} \]

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

\[ \text{Value} = \frac{86 \div 2}{24 \div 2} = \frac{43}{12} \]

Thus, the value of the given trigonometric expression is \(\frac{43}{12}\).

Revision Table: Key Trigonometric Values

Additional Information: Trigonometric Identities Used

This problem utilised the following concepts:

• Values of trigonometric functions for standard angles (\(30^\circ, 45^\circ, 60^\circ, 90^\circ\)).
• Properties of exponents with trigonometric functions, e.g., \(\sin^2 \theta = (\sin \theta)^2\).
• Arithmetic operations with fractions.
• Implicitly, the angle subtraction formula for sine: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). While the denominator calculation can be done by substituting values directly, recognizing this identity provides an alternative method and reinforces understanding of trigonometric formulas.