\( \left. \begin{array} { l } { 2.7 \frac { \sin ( 450 ^ { \circ } - x ) \cdot \tan ( x - 180 ^ { \circ } ) \sin 23 ^ { \circ } \cos 23 ^ { \circ } } { \cos 44 ^ { \circ } \cdot \sin ( - x ) } } \\ { 2.8 \frac { \sin 130 ^ { \circ } \cdot \tan 60 ^ { \circ } } { \cos 540 ^ { \circ } \cdot \tan 230 ^ { \circ } \cdot \sin 400 ^ { \circ } } } \\ { 2.9 \quad \frac { ( 1 - \sqrt { 2 } \sin 75 ^ { \circ } ) ( \sqrt { 2 } \sin 75 ^ { \circ } + 1 ) } { \sin ( 360 ^ { \circ } - x ) \cdot \sin ( 90 ^ { \circ } + x ) } } \\ { 2.10 } \end{array} \right. \)

We shall simplify the given expressions one‐by‐one. (Note: In the list the item “2.10” does not show any expression to simplify.) ───────────────────────────── 2.7.   Expression:     [sin(450° – x) · tan(x – 180°) · sin 23° · cos 23°]⁄[cos 44° · sin(–x)] Step 1. Write each part in a simpler form.  • sin(450° – x): Notice that 450° = 360° + 90°, so   sin(450° – x) = sin[(360° + 90°) – x] = sin(90° – x) = cos x.  • tan(x – 180°): Since the tangent is 180°‐periodic,   tan(x – 180°) = tan x.  • sin(–x) = – sin x.  • Also recall that sin 23° · cos 23° = (1/2) sin(46°) by the double‐angle formula.  • Finally, note that cos 44° = sin(90° – 44°) = sin(46°). Step 2. Substitute these into the expression:   = [cos x · tan x · sin 23° · cos 23°]⁄[cos 44° · (– sin x)]   = [cos x · (sin x⁄cos x) · sin 23° · cos 23°]⁄[– sin x · cos 44°]   = [sin x · sin 23° · cos 23°]⁄[– sin x · cos 44°]. Step 3. Cancel common factor sin x (assuming sin x ≠ 0):   = [sin 23° · cos 23°]⁄[– cos 44°]. Step 4. Replace the product and the cosine:   = [(1/2) sin 46°]⁄[– sin 46°] = –1/2. Thus, the simplified result for 2.7 is –½. ───────────────────────────── 2.8.   Expression:     [sin 130° · tan 60°]⁄[cos 540° · tan 230° · sin 400°]. Step 1. Rewrite each trigonometric function.  • sin 130°: Since 130° = 180° – 50°, we have sin 130° = sin 50°.  • tan 60° = √3.  • cos 540°: Notice that 540° = 360° + 180° so cos 540° = cos 180° = –1.  • tan 230°: Write 230° = 180° + 50° and use the 180°–periodicity of tangent, so tan 230° = tan 50°.  • sin 400°: Since 400° = 360° + 40°, sin 400° = sin 40°. Step 2. Substitute these:   = [sin 50° · √3]⁄[ (–1) · tan 50° · sin 40°]. Step 3. Write tan 50° as sin 50°⁄cos 50°:   = [sin 50° √3]⁄[– (sin 50°⁄cos 50°) · sin 40°]   = [sin 50° √3 · cos 50°]⁄[– sin 50° · sin 40°]. Step 4. Cancel sin 50° (provided it is nonzero):   = [√3 · cos 50°]⁄[– sin 40°]. Step 5. Notice that cos 50° = sin(90° – 50°) = sin 40°:   = [√3 · sin 40°]⁄[– sin 40°] = –√3. Thus, the simplified result for 2.8 is –√3. ───────────────────────────── 2.9.   Expression:     [(1 – √2 sin 75°)(√2 sin 75° + 1)]⁄[sin(360° – x) · sin(90° + x)]. Step 1. Simplify the numerator. Write it as a product of two conjugates:  Let A = √2 sin 75°. Then   (1 – A)(1 + A) = 1 – A² = 1 – 2 sin² 75°. Step 2. Simplify the denominator.  • sin(360° – x): Using the identity sin(360° – x) = – sin x.  • sin(90° + x) = cos x    (since sin(90° + x) = cos x). Thus, the denominator is (– sin x · cos x) = – (sin x cos x). Step 3. So the expression becomes:   = [1 – 2 sin² 75°]⁄[– sin x cos x]. Step 4. Recognize that 1 – 2 sin²θ = cos 2θ. With θ = 75°, we have:   1 – 2 sin² 75° = cos 150°.  But cos 150° = – cos 30° = – (√3⁄2). Thus, the numerator is –(√3⁄2). Substitute it in:   = [– (√3⁄2)]⁄[– sin x cos x] = (√3⁄2)⁄(sin x cos x). Step 5. Recognize that 2 sin x cos x = sin 2x. Therefore,   (sin x cos x) = (1/2) sin 2x, so   (√3⁄2)/(sin x cos x) = (√3⁄2)/((1/2) sin 2x) = (√3)/(sin 2x). Thus, the simplified result for 2.9 is (√3)/(sin 2x). ───────────────────────────── 2.10.  No expression is given for 2.10 in the problem statement. ───────────────────────────── Summary of Answers:  2.7: –½  2.8: –√3  2.9: (√3)/(sin 2x)  2.10: (No expression provided) These are acceptable simplified forms for the given expressions.