Question

5.simplify the following expression using exact values. 3 marks
cos 60°(1 - sin 30°) + 4 tan 45°(tan 60° - sin 45°)

Explanation:

Step1: Substitute trigonometric values

We know that \(\cos60^{\circ}=\frac{1}{2}\), \(\sin30^{\circ}=\frac{1}{2}\), \(\tan45^{\circ} = 1\), \(\tan60^{\circ}=\sqrt{3}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\).
Substitute these values into the expression:

$$
LATEXBLOCK0
$$

Step2: Simplify the first - part of the expression

Simplify \(\frac{1}{2}(1 - \frac{1}{2})\):

$$ \frac{1}{2}(1-\frac{1}{2})=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4} $$

Step3: Simplify the second - part of the expression

Simplify \(4\times1\times(\sqrt{3}-\frac{\sqrt{2}}{2})\):

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

Step4: Combine the two parts

Combine \(\frac{1}{4}\) and \(4\sqrt{3}-2\sqrt{2}\):

$$ \frac{1}{4}+4\sqrt{3}-2\sqrt{2}=4\sqrt{3}-2\sqrt{2}+\frac{1}{4} $$

Answer:

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