Let's simplify each of the expressions step-by-step.
### Part (a)
Simplify [tex]\(\frac{1}{2} \sin x \cos x\)[/tex]:
1. Use the double angle identity for sine: [tex]\(\sin(2x) = 2 \sin x \cos x\)[/tex]
2. Substitute into the expression: [tex]\(\frac{1}{2} \sin x \cos x = \frac{1}{2} \cdot \frac{\sin(2x)}{2} = \frac{1}{4} \sin(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \frac{1}{2} \sin x \cos x = 0.25 \sin(2x) \][/tex]
### Part (b)
Simplify [tex]\(\sin^2 x - \cos^2 x\)[/tex]:
1. Use the Pythagorean identity: [tex]\(\sin^2 x = 1 - \cos^2 x\)[/tex]
2. Substitute into the expression: [tex]\((1 - \cos^2 x) - \cos^2 x = 1 - 2 \cos^2 x\)[/tex]
3. Recognize that [tex]\(1 - 2 \cos^2 x = - \cos(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \sin^2 x - \cos^2 x = -\cos(2x) \][/tex]
### Part (c)
Simplify [tex]\(\frac{\tan x}{1 - \tan^2 x}\)[/tex]:
1. Use the tangent identity: [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex] and [tex]\(\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}\)[/tex]
2. Rearrange the terms: [tex]\(\frac{\tan x}{1 - \tan^2 x} = \frac{1}{2} \tan(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \frac{\tan x}{1 - \tan^2 x} = \frac{\tan(2x)}{2} \][/tex]
### Part (d)
Simplify [tex]\(\frac{\sin^4 x - \cos^4 x}{\cos(2x)}\)[/tex]:
1. Factor the numerator: [tex]\(\sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)\)[/tex]
2. Recognize [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]
3. Thus, [tex]\(\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x\)[/tex]
4. From part (b), [tex]\(\sin^2 x - \cos^2 x = -\cos(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \frac{\sin^4 x - \cos^4 x}{\cos(2x)} = -1 \][/tex]
### Part (e)
Simplify [tex]\(\frac{4 \sin^3 x - 2 \sin x}{\cos x \cos(2x)}\)[/tex]:
1. Factor out [tex]\(2 \sin x\)[/tex] from the numerator: [tex]\(4 \sin^3 x - 2 \sin x = 2 \sin x (2 \sin^2 x - 1)\)[/tex]
2. Use the double angle identity for cosine: [tex]\(2 \sin^2 x - 1 = - \cos(2x)\)[/tex]
3. Thus, [tex]\(4 \sin^3 x - 2 \sin x = 2 \sin x \cdot (-\cos(2x)) = -2 \sin x \cos(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \frac{4 \sin^3 x - 2 \sin x}{\cos x \cos(2x)} = -2 \tan x \][/tex]
### Part (f)
Simplify [tex]\(\frac{4 \sin^2 x - 4 \sin^4 x}{\sin(2x)}\)[/tex]:
1. Factor out [tex]\(4 \sin^2 x\)[/tex] from the numerator: [tex]\(4 \sin^2 x - 4 \sin^4 x = 4 \sin^2 x (1 - \sin^2 x)\)[/tex]
2. Use the Pythagorean identity: [tex]\(1 - \sin^2 x = \cos^2 x\)[/tex]
3. Thus, [tex]\(4 \sin^2 x - 4 \sin^4 x = 4 \sin^2 x \cos^2 x\)[/tex]
4. Use the double angle identity for sine: [tex]\(2 \sin x \cos x = \sin(2x)\)[/tex]
So, the simplified form is:
[tex]\[ \frac{4 \sin^2 x - 4 \sin^4 x}{\sin(2x)} = \sin(2x)\][/tex]
