Let = r, (cos 0, + isin 0, ) and r, (cos 0, + i sin 0, ). Z, = X1 + iy, 1 Z 2 X2 + iy, Then show that (i): r," .r," (cos n (0, + 0, ) + i sin n (0, + 0 , )) n z"1.Z 1 1 (ii): (cosn (0, - 0, ) + i sin n (0, - 0, )) %3D 2 n 2

Q: 20 a Show that sin (A + B) + sin (A – B) = 2 sin A cos B. b Hence find: i 2 sin 3x cos 2x dx ii sin…

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Q: Do the functions y, = cos4t and y, = sin4t constitute a fundamer %3D The functions constitute a…

A: The given differential equation is y''+4y=0 the auxiliary equation is D2+4=0 D2=-4D=±2i therefore…

Q: Let F = Vowhere o = e cos (Vy ). If C is the cardioid defined by ((2+ sin 0) sin 0, (2 + sin 0) cos…

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Q: If U(x, y) =-e Y sin x and f(z) = U(x, y) + iV(x, y) is analytic, then V(x, y) = %3D -e cos x e cosx…

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Q: Given that Y1 = x-1/2 cos x and y2 = x-1/2 sin x from a fundamental set of solutions of (--)- x²y" +…

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Q: 1. If y= tan x, prove that y"= 2(y² +1)(3y² +1).

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Q: snip

A: Given, cosjx,coskx, ∋j≠k, k≥1 if inner product of two function f(x), g(x) on the interval [a,b]…

Q: 1. If y= tan x, prove that y``= 2(y² +1)(3y² +1).

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Q: 3. u, = u, - sin x cos t, 1 >0 В.С.: и(0, г) %3D0, = () 1.С.: «(х, 0) %3D 0, u,(x. 0) = 0

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Q: 2. Let f(r, y,z) = In(r² + 2) + ye* – cos(2yz). Find faya- COS

A: see below the answer as per bartleby guideline we have to solve the first question onl

Q: Suppose f(x, y) : where x = r cos 0 and y = r sin 0. 4 af af and ar Find when r 1, 0 -

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Q: Q1: show that1) d/dz [cosz] = -sin z, 2) 1+tan2 z = sec2 z

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Q: Let a, b be real numbers and consider the integral (ax² + b) cos(x)dx. Using integration by parts…

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Q: Z1 "(cos(0, r2 Z2 02) + isin(0, – 02)](z2 # 0). 85. Let z1 = ri(cos 0, + isin 01) and z2 = r2(cos 0,…

A: Given z1=r1cosθ1+isinθ1 and z2=r2cosθ2+isinθ2.

Q: If F(x,y,z)=sin(x³y²)+cos (Vxy2)+x®y²z, then fyz(1,-2,1)=

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Q: Find the residues of sin(2²) at z = 0 and z = π. (sin z)2

A: To find the residue of sinz2sinz2 at z=0 and z=π Let  fz=sinz2sinz2=sinz2csc2z…

Q: Let f(x, y, z) = ln(x² + z) + y²e"z – cos(2yz). Find fzya- CoS %3D

A: use partial differentiation for finding the derivative.

Q: التاريخ اليوم موضوع الدرس A) Discuss the conrergence of facx) %3D on A = Co,0)

A: Convergence of the sequence

Q: A force of F(x)=x2−cos(3x)+2, x is in meters, acts on an object. What is the work required to move…

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Q: Suppose that w = a7 · exp(2y) · cos(4z) with z- in (+ 5), x = sin (t+ y = In (t +2), z = t.

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Q: 1. z, =1-j2 and Z,=2+j3; Evaluate cos[Z,J?

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Q1: show that1) d/dz [cosz] = -sin z , 2) 1+tan2 z = sec2 z *

A: We have to use derivative definition.

Q: Let f(x, y) = x² + y² and r(t) = (cos 1, sin t). (a) Find f(r(1)) without making any calculations.…

A: a) Equation of the circle with the centre at the origin that is (0,0) is x2+y2=a2 then every point…

Q: Compute | cos z dz. cos z +C 3 cos a + sin z +C 3 sin r + cos x +C 3 Cos x - sin I sin z + 5 sin x…

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Q: Hi, there are two questions a) and b) attached  Thanks so Much!!!!

A: (a)consider the given complex number:

Q: Q1: show that1) d/dz [cosz] = -sin z , 2) * 1+tan2 z = sec2 z

A: In both solutions i have used the definition

Q: 5. If z = f(x, y), where x = r cos 0 and r sin 0, find z and and hence show that: (). (-(( 2 2 dz dz…

A: To find the partial derivatives of the following function in two variables by chain rule.

Q: The area of the region inside r = 2+ cos 0 and outside r = 4 – 3 cos 0 is given by | ((2+ cos 0)² –…

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Q: f(x) = In(2 + cos x). -2.125, xk+1 = Ik +0.5, yk = f (xk), k E N To = %3D %3D = 0r

A: The function is fx=ln2+cosx Conditions: x0=-2.125, xk+1=xk+0.5, yk=fxk, k∈ℕ xk+1=xk+0.5…

Q: 2. Let P:=-<-<0<, be a partition of [-], and (sin(2r), 0≤x≤fi f(x)=sin(2x), -*<0; (cos(2x) -1<x<-

A: The information of your question is complete, though, you have nowhere mentioned what exactly you…

Q: 2-If w(x, t) = Sin(x - + 3t) + Cos(2x+ 6t), prove that wtt = 9wxx %3D %3D

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Q: Suppose f(x, y) = x2 + y? where x = r cos 0 and y = r sin 0. af and Or 1, 0 = 6 Find -when r

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Q: 3. Given that z = e" cos (), u = 4e2" – sin 2y, v = e" + n tan r. COS - Use chain rule to find dz/ðx…

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Q: Use Cramer's rule to solve for x' and y' in terms of x and y. x = x' cos 0 – y' sin 0 y = x' sin 0 +…

A: Solution is given as below:

Q: (a) If u(x, y) = cos-1*" + y* ) x² + y? = COS then prove that ди du + y = -2cotu

A: Given: ux,y=cos-1x4+y4x2+y2

Q: a Show that (sin(m + n)x + sin(m – n)x) b Hence, without using the product rule, differentiate sin…

A: We will find out the required values.

Q: (a) Prove that dv 1 2 u? + v2 arctan u- + C. 2 u 2, /2/2 ru (b) Prove that I dv du 2 - u? + v2 18 by…

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Q: Compute Cos (x + k ) cos a dx by using substitution technique where k is a constant. [Hint: cos(A±…

A: The given integral is cos(x+k)/cosx.

Q: -1 If cos x+cos y+cos z= T, prove that x +y +z +2xyz = 1

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Q: Q1: show that1) d/dz [cosz] = -sin z , 2) 1+tan2 z = sec2 z

A: (1) Use the first principle to derive the derivative of cos(z),…

Q: Find the Wronskian for y, = sin 3x , and y2 = cos 3x.

A: The Wronskian, W, of two functions, f and g is calculated as follows. W=fgf'g' , where, f' , g'are…

Q: Let f:R^2 --> R^3 be given by f(x1,x2)=(e^(x1+x2^2, x2+cos(x1,x2), x1^2+x2^2+2) Compute the…

A: Using jacobian matrix,

Q: Let f(r, y, z) = In(x² + 2) + y?e - cos(2y2). Find feyr-.

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Q: Prove that: Z,Z2 =r,r2[cos(0, + 02) + i sin(0, + 02)] 1.

A: NOTE- As we are allowed to answer one question in one session so, i have answered first one for you.…

Q: Let z=In(x²+y²) where x=ue"sin t and y = ue"cos t. If u=- 2 and 1=0 then is equal to %3D ....

A: Multivariable function: When a function f has more than one independent variable and only one…

Q: 2 tan-1(y). x In(xyz), q ƏV when x = y = % = 1. ду Let V = eP cos(qr), with p = z cos(TY) and r = By…

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Q: . Let z = x2 – x*y where x = r sin 0 and y = r cos 0. az and ae az Use the chain rule to find ar…

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Q: Let = (sin x cos y, cos I sin y, In z).- Compute div i %3D

A: Given,          v→ =  sinxcosy , cosxsiny , lnz

Q: Integrate the trig identity 2 cos j x cos kx = cos(j + k )x + cos(j -k )x to show that cos j x is…

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