Evaluating Trigonometric Expressions at Standard Angles

To find the value of the given trigonometric expression, we need to substitute the standard trigonometric values for the angles 30°, 60°, 45°, and 90°. Let's list the required standard values:

The given expression is:

\(\frac{2 \sin^2 30° \tan 60°-3 \cos^2 60° \sec^2 30°}{4\cot^2 45°-\sec^2 60°+ \sin^2 60°+\cos^2 90°}\)

Evaluate the Numerator

The numerator is \(2 \sin^2 30° \tan 60°-3 \cos^2 60° \sec^2 30°\). Substitute the standard values:

• \(\sin 30° = \frac{1}{2}\)
• \(\tan 60° = \sqrt{3}\)
• \(\cos 60° = \frac{1}{2}\)
• \(\sec 30° = \frac{2}{\sqrt{3}}\)

Substituting these values into the numerator:

\(2 \left(\frac{1}{2}\right)^2 (\sqrt{3}) - 3 \left(\frac{1}{2}\right)^2 \left(\frac{2}{\sqrt{3}}\right)^2\)

Calculate the squares:

\(2 \left(\frac{1}{4}\right) (\sqrt{3}) - 3 \left(\frac{1}{4}\right) \left(\frac{4}{3}\right)\)

Simplify the terms:

\(\frac{2\sqrt{3}}{4} - 3 \left(\frac{4}{12}\right)\)

\(\frac{\sqrt{3}}{2} - 3 \left(\frac{1}{3}\right)\)

\(\frac{\sqrt{3}}{2} - 1\)

Combine into a single fraction:

\(\frac{\sqrt{3} - 2}{2}\)

So, the numerator evaluates to \(\frac{\sqrt{3} - 2}{2}\).

Evaluate the Denominator

The denominator is \(4\cot^2 45°-\sec^2 60°+ \sin^2 60°+\cos^2 90°\). Substitute the standard values:

• \(\cot 45° = 1\)
• \(\sec 60° = 2\)
• \(\sin 60° = \frac{\sqrt{3}}{2}\)
• \(\cos 90° = 0\)

Substituting these values into the denominator:

\(4(1)^2 - (2)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 + (0)^2\)

Calculate the squares:

\(4(1) - 4 + \frac{3}{4} + 0\)

Simplify the terms:

\(4 - 4 + \frac{3}{4} + 0\)

\(0 + \frac{3}{4}\)

\(\frac{3}{4}\)

So, the denominator evaluates to \(\frac{3}{4}\).

Calculate the Final Value

Now, divide the numerator by the denominator:

Value of expression = \(\frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{\sqrt{3} - 2}{2}}{\frac{3}{4}}\)

To divide by a fraction, multiply by its reciprocal:

\(\frac{\sqrt{3} - 2}{2} \times \frac{4}{3}\)

Multiply the fractions:

\(\frac{(\sqrt{3} - 2) \times 4}{2 \times 3}\)

\(\frac{4(\sqrt{3} - 2)}{6}\)

Simplify the fraction by dividing the numerator and denominator by 2:

\(\frac{2(\sqrt{3} - 2)}{3}\)

This is the final value of the expression.

Comparing with Options

Let's compare the calculated value \(\frac{2(\sqrt{3} - 2)}{3}\) with the given options:

1. \(\frac{1}{30}{ (\sqrt3 - 2)}\)
2. \(\frac{2(\sqrt 3 + 2)}{3}\)
3. \(\frac{2(\sqrt 3 - 2)}{3}\)
4. \(\frac{1}{30}{ (\sqrt3 + 2)}\)

The calculated value matches option 3.

Revision Table: Key Trigonometric Values

Additional Information: Trigonometric Identities

This problem relies on understanding standard trigonometric values. It's also useful to remember fundamental trigonometric identities that can simplify expressions:

• Reciprocal Identities:
  • \(\csc \theta = \frac{1}{\sin \theta}\)
  • \(\sec \theta = \frac{1}{\cos \theta}\)
  • \(\cot \theta = \frac{1}{\tan \theta}\)
• Quotient Identities:
  • \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
  • \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
• Pythagorean Identities:
  • \(\sin^2 \theta + \cos^2 \theta = 1\)
  • \(\tan^2 \theta + 1 = \sec^2 \theta\)
  • \(1 + \cot^2 \theta = \csc^2 \theta\)

These identities help in simplifying complex trigonometric expressions, although the problem here mainly required knowing the values at specific angles.