Activity 2 Simplify the following WITHOUT the use of a calculator: \( 2.1 \frac{\tan \left(180^{\circ}+x\right) \cdot \cos \left(360^{\circ}-x\right)}{\sin (180-x) \cdot \cos \left(90^{\circ}+x\right)+\cos \left(540^{\circ}+x\right) \cdot \cos (-x)} \)

We wish to simplify the expression   2.1 · [tan(180° + x) · cos(360° – x)]     ————————————————————————————————————     [sin(180° – x) · cos(90° + x) + cos(540° + x) · cos(–x)] Step 1. Simplify the numerator. • Recall that tan(θ + 180°) = tan θ. Thus,   tan(180° + x) = tan x. • Cosine has period 360°. Also, cos(θ) is even so cos(–x) = cos x. Therefore,   cos(360° – x) = cos(–x) = cos x. So the numerator becomes:   tan x · cos x. Step 2. Simplify the denominator. The denominator has two parts. We simplify each term: Term A: sin(180° – x) · cos(90° + x)   • sin(180° – x) = sin x.   • cos(90° + x) = – sin x     (since cos(90° + θ) = – sin θ). Thus, Term A = sin x · (– sin x) = – sin² x. Term B: cos(540° + x) · cos(–x)   • Notice that 540° = 360° + 180°. Using the periodicity of cosine,     cos(540° + x) = cos(180° + x) = – cos x     (because cos(180° + θ) = – cos θ).   • Also, cos(–x) = cos x. Thus, Term B = (– cos x) · cos x = – cos² x. Adding both terms in the denominator:   Denom = – sin² x – cos² x = – (sin² x + cos² x). Since sin² x + cos² x = 1, we have   Denom = –1. Step 3. Write the simplified expression. The original expression becomes:   2.1 · (tan x · cos x) / (–1) = –2.1 · (tan x · cos x). Step 4. Final simplification. Recall tan x = sin x / cos x. Then:   tan x · cos x = (sin x / cos x) · cos x = sin x. Thus, the entire expression simplifies to:   –2.1 · sin x. Answer: –2.1 sin x.