(4) \( \frac{2 \tan \left(x+180^{\circ}\right) \cos \left(720^{\circ}-x\right)}{\cos \left(450^{\circ}+x\right)}+1 \) (5) \( \frac{\cos \left(90^{\circ}-2 x\right)}{2-2 \sin (-x) \cos \left(90^{\circ}+x\right)} \) (6) \( \frac{\cos ^{2}\left(180^{\circ}-\theta\right)-\cos ^{2} \alpha-\sin ^{2}\left(\alpha-180^{\circ}\right)}{\cos \left(360^{\circ}-2 \theta\right)-1} \)

We will simplify each expression step‐by‐step. ───────────────────────────── (4) Expression:   [2 · tan(x + 180°) · cos(720° – x)] / cos(450° + x) + 1 Step 1. Use periodicity:  • tan(x + 180°) = tan x   (since the period of tan is 180°)  • 720° = 2·360°, so cos(720° – x) = cos(–x) = cos x   (cosine is an even function) Now the expression becomes:   [2 · tan x · cos x] / cos(450° + x) + 1 Step 2. Recall that tan x = sin x/cos x; multiplying by cos x gives:   tan x · cos x = sin x Thus, the expression is now:   [2 sin x] / cos(450° + x) + 1 Step 3. Simplify cos(450° + x):   450° = 360° + 90° so,   cos(450° + x) = cos(360° + 90° + x) = cos(90° + x)   But cos(90° + x) = – sin x Now our expression is:   [2 sin x] / (– sin x) + 1 = –2 + 1 = –1 ───────────────────────────── (5) Expression:   cos(90° – 2x) / [2 – 2 sin(–x) cos(90° + x)] Step 1. Simplify the numerator:   cos(90° – 2x) = sin(2x)   (since cos(90° – θ) = sin θ) Step 2. Simplify the denominator:  • sin(–x) = – sin x   (sine is odd)  • cos(90° + x) = – sin x   (since cos(90° + x) = – sin x) Multiplying these:   sin(–x) cos(90° + x) = (– sin x)(– sin x) = sin² x Then the denominator becomes:   2 – 2 sin² x = 2(1 – sin² x) = 2 cos² x    (using sin² x + cos² x = 1) Step 3. Now the expression is:   sin(2x) / (2 cos² x) Recall that:   sin(2x) = 2 sin x cos x So,   [2 sin x cos x] / (2 cos² x) = sin x / cos x = tan x ───────────────────────────── (6) Expression:   [cos²(180° – θ) – cos²α – sin²(α – 180°)] / [cos(360° – 2θ) – 1] Step 1. Simplify the numerator term-by-term.  • cos(180° – θ) = – cos θ   (cosine identity), hence:   cos²(180° – θ) = cos² θ  • For sin²(α – 180°):   sin(α – 180°) = – sin(180° – α)   (sine is odd)   But sin(180° – α) = sin α, so   sin(α – 180°) = – sin α,   and thus sin²(α – 180°) = sin² α Now the numerator is:   cos² θ – cos²α – sin²α But note that:   cos²α + sin²α = 1 So, the numerator becomes:   cos² θ – 1 = – sin² θ    (since cos² θ – 1 = – sin² θ) Step 2. Simplify the denominator.   cos(360° – 2θ) = cos 2θ   (cosine is periodic with period 360°) Thus, the denominator is:   cos 2θ – 1 Recall the double-angle identity:   cos 2θ – 1 = – 2 sin² θ Step 3. Now, the entire expression becomes:   (– sin² θ) / (– 2 sin² θ) = 1/2 ───────────────────────────── Final Answers:  (4) simplifies to –1  (5) simplifies to tan x  (6) simplifies to ½