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: Understanding the Concept:
This problem requires substituting standard trigonometric values into an expression and simplifying. We also use the identity \(\sin^2 \theta + \cos^2 \theta = 1\).
Step 2: Key Formula or Approach:
Use: \(\cos 60^\circ = \frac{1}{2}\), \(\sec 30^\circ = \frac{2}{\sqrt{3}}\), \(\tan 45^\circ = 1\), and \(\sin^2 \theta + \cos^2 \theta = 1\).
Step 3: Detailed Explanation:
First, observe the denominator:
\[ \sin^2 30^\circ + \cos^2 30^\circ = 1 \]
Now, simplify the numerator:
\[ 5 \left(\frac{1}{2}\right)^2 + 4 \left(\frac{2}{\sqrt{3}}\right)^2 - (1)^2 \]
\[ = 5 \left(\frac{1}{4}\right) + 4 \left(\frac{4}{3}\right) - 1 \]
\[ = \frac{5}{4} + \frac{16}{3} - 1 \]
Find the LCM of denominators (4 and 3) which is 12:
\[ = \frac{5 \times 3 + 16 \times 4 - 1 \times 12}{12} \]
\[ = \frac{15 + 64 - 12}{12} \]
\[ = \frac{79 - 12}{12} = \frac{67}{12} \]
Step 4: Final Answer:
The evaluated value is \(\frac{67}{12}\).