Question

Evaluate: \(\frac{5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}\)

Hint:

Always look for the identity \(\sin^2 \theta + \cos^2 \theta = 1\) first; it usually simplifies the denominator to 1, saving a lot of complex calculation time.

Answer 1

Step 1: Reconsidering the Problem Setup:
The given expression involves trigonometric values for angles and requires simplification. We will use a step-by-step approach to first tackle the numerator and then handle the denominator using standard trigonometric identities. 
Step 2: Trigonometric Identities to Apply:
- \(\cos 60^\circ = \frac{1}{2}\) - \(\sec 30^\circ = \frac{2}{\sqrt{3}}\) - \(\tan 45^\circ = 1\) - Fundamental Identity: \(\sin^2 \theta + \cos^2 \theta = 1\) 
Step 3: Breaking Down the Expression:
Let's focus on simplifying the numerator and denominator separately: 
Denominator: Using the identity for \( \sin^2 30^\circ + \cos^2 30^\circ \): \[ \sin^2 30^\circ + \cos^2 30^\circ = 1 \] This simplification reduces the denominator to 1.

Numerator: Now simplify the terms in the numerator: \[ 5 \left(\frac{1}{2}\right)^2 + 4 \left(\frac{2}{\sqrt{3}}\right)^2 - (1)^2 \] First calculate each part: \[ = 5 \times \frac{1}{4} + 4 \times \frac{4}{3} - 1 \] \[ = \frac{5}{4} + \frac{16}{3} - 1 \] To combine these terms, find the least common denominator (LCD), which is 12: \[ = \frac{5 \times 3}{12} + \frac{16 \times 4}{12} - \frac{1 \times 12}{12} \] \[ = \frac{15}{12} + \frac{64}{12} - \frac{12}{12} \] Combine the terms in the numerator: \[ = \frac{15 + 64 - 12}{12} \] \[ = \frac{67}{12} \] Step 4: Final Answer:
Thus, the value of the expression is: \[ \frac{67}{12} \]