answer:Consider the trigonometric expression {eq}sin (x-y) sin y+cos (x-y) cos y{/eq}. Apply the difference formulas {eq}sin (x-y)=sin x cos y-cos x sin y{/eq} and {eq}cos (x-y)=cos x cos y+sin x sin y{/eq}. {eq}begin{aligned} sin (x-y) sin y+cos (x-y) cos y &=(sin x cos y-cos x sin y) sin y+(cos x cos y+sin x sin y)cos y &=sin x sin y cos y-cos x sin^2 y+cos x cos^2 y+sin x sin y cos y &=2sin x sin y cos y+cos x (cos^2 y-sin^2 y) end{aligned}{/eq} Use the double angle formula {eq}cos 2x=cos^2 x-sin^2 x{/eq}. {eq}begin{aligned} sin (x-y) sin y+cos (x-y) cos y &=2sin x sin y cos y+cos x (cos 2y) &=sin 2y cos x + cos x cos 2y &=cos x (sin 2y + cos 2y) end{aligned}{/eq} Therefore, the simplified form of the expression is {eq}mathbf{cos x (sin 2y + cos 2y)}{/eq}.

answer:The first-order Taylor expansion of f(g(x)) about x = -4 can be found by applying the chain rule and the linear approximation of the functions f(x) and g(x) at x = -4. The Taylor series of f(x) = sin(x) around x = -4 is sin(-4) + (x+4)cos(-4), and the Taylor series of g(x) = x^2 around x = -4 is (-4)^2 + 2(x+4)(-4) = 16 - 8(x+4). Now, we substitute g(x) into f(x): f(g(x)) = sin(g(x)) = sin(16 - 8(x+4)) Applying the first-order approximation of sin(y) around y = 16, where y = 16 - 8(x+4): sin(16 - 8(x+4)) approx sin(16) - 8(x+4)cos(16) Expanding and simplifying, we get: sin(16) - 8(x+4)cos(16) = -8(x+4)cos(16) + sin(16) Now, we rewrite the expression in the form of a linear approximation: -8(x+4)cos(16) + sin(16) = -(8cos(16) + sin(16)) + (x+4)(8cos(16)) Rearranging terms, we have the first-order expansion: (x+4) (8cos(16)) - 8cos(16) + sin(16) Simplifying further: 8cos(16)(x+4) - 8cos(16) + sin(16) - sin(16) The sin(16) terms cancel out: 8cos(16)(x+4) - 8cos(16) Simplifying again, we get: 8cos(16)(x+4 - 1) Finally, this simplifies to: 8cos(16)(x + 3) So the first-order expansion of f(g(x)) about x = -4 is 8cos(16)(x + 3).

answer:The simplest example of a unitary ring with exactly one invertible element is the trivial ring, also known as the zero ring, denoted by {0}. In this ring, the only element is 0, and since 0 multiplied by 0 is still 0, the element 0 is its own multiplicative identity (the unity in a unitary ring). As there are no other elements in the ring, 0 is the only invertible element, as it is the only one that has an inverse (itself).

answer:In axiomatic set theory, a "set" is a well-defined object that can be described within the formal theory. A "collection," on the other hand, is a more intuitive notion that may not be definable within the theory. The domain of discourse is a collection of objects that the theory is concerned with, but it may not be a set within the theory itself. For example, in Zermelo–Fraenkel set theory (ZFC), the collection of all ZFC-sets is not a ZFC-set due to Russell's paradox. However, it can be considered a "class" or an object in a different theory, such as Neumann–Bernays–Gödel set theory (NBG).