Evaluating the Trigonometric Expression

The question asks us to evaluate the expression: $$ \frac{5\cos^2 120^\circ + 4\sec^2 30^\circ - \tan^2 135^\circ} {\sin^2 30^\circ + \cos^2 30^\circ} $$ To solve this, we need to find the values of the trigonometric functions at the given angles and substitute them into the expression.

Step 1: Determine Trigonometric Values

First, let's find the values for each trigonometric function required:

• Cosine of 120 degrees: $\cos 120^\circ = -\frac{1}{2}$. Therefore, $\cos^2 120^\circ = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}$.
• Secant of 30 degrees: $\sec 30^\circ = \frac{1}{\cos 30^\circ}$. Since $\cos 30^\circ = \frac{\sqrt{3}}{2}$, we have $\sec 30^\circ = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$. Therefore, $\sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}$.
• Tangent of 135 degrees: $\tan 135^\circ = -1$. Therefore, $\tan^2 135^\circ = (-1)^2 = 1$.
• Sine of 30 degrees: $\sin 30^\circ = \frac{1}{2}$. Therefore, $\sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.
• Cosine of 30 degrees: $\cos 30^\circ = \frac{\sqrt{3}}{2}$. Therefore, $\cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$.

Step 2: Substitute Values into the Expression

Now substitute these squared values back into the original expression:

Step 3: Simplify the Numerator

Let's simplify the numerator first:

Numerator = $5 \times \frac{1}{4} + 4 \times \frac{4}{3} - 1$

Numerator = $\frac{5}{4} + \frac{16}{3} - 1$

To add these fractions, we find a common denominator, which is 12:

Numerator = $\frac{5 \times 3}{4 \times 3} + \frac{16 \times 4}{3 \times 4} - \frac{1 \times 12}{1 \times 12}$

Numerator = $\frac{15}{12} + \frac{64}{12} - \frac{12}{12}$

Numerator = $\frac{15 + 64 - 12}{12}$

Numerator = $\frac{79 - 12}{12}$

Numerator = $\frac{67}{12}$

Step 4: Simplify the Denominator

Now, let's simplify the denominator:

Denominator = $\sin^2 30^\circ + \cos^2 30^\circ$

Using the trigonometric identity $\sin^2 \theta + \cos^2 \theta = 1$, we know that:

Denominator = $\frac{1}{4} + \frac{3}{4} = \frac{1+3}{4} = \frac{4}{4} = 1$.

Alternatively, substituting the calculated values:

Denominator = $\frac{1}{4} + \frac{3}{4} = 1$.

Step 5: Calculate the Final Value

Now, divide the simplified numerator by the simplified denominator:

Thus, the value of the expression is $\frac{67}{12}$.