Question

Evaluate $-5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan 45^\circ$

Hint:

When simplifying trigonometric expressions, remember to use standard values for common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\).

Answer 1

Step 1: Simplify each trigonometric expression.
We are asked to evaluate the expression: \[ -5 \cos^2 60^\circ + 4 \sec^2 30^\circ - \tan 45^\circ \] We know the following values:
- \(\cos 60^\circ = \frac{1}{2}\), so \(\cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(\sec 30^\circ = \frac{2}{\sqrt{3}}\), so \(\sec^2 30^\circ = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}\)
- \(\tan 45^\circ = 1\)

Step 2: Substitute the values into the expression.
Substitute the values into the given expression: \[ -5 \cdot \frac{1}{4} + 4 \cdot \frac{4}{3} - 1 \] \[ = -\frac{5}{4} + \frac{16}{3} - 1 \]
Step 3: Simplify the expression.
To simplify, first find a common denominator: \[ -\frac{5}{4} + \frac{16}{3} - 1 = -\frac{5}{4} + \frac{16}{3} - \frac{4}{4} \] \[ = \frac{-5 + 64 - 16}{12} = \frac{43}{12} \] Thus, the value of the expression is: \[ \frac{43}{12} \]