Prove that: Z, Z2 =r,r2[cos(8, + 02) + i sin(0, + 02)] 2=cos(0, - 62) + i sin(0, – 02)] Z2 3. (cos 0 + i sin 0)" = cos no + i sin ne

Q: (a) cos(iz) = cos(iz) (b) sin(iz) = sin (iz) for all z; if and only if z = nлi (n = 0, +1, +2, ...).

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Q: 2. If z = cos 0 + i sin 0 where 0 is real, show that 1 1 - i tan- = - 1+z 2z 1-z2 Express: а) 1+z2…

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Q: Find the product of zı and z2 Given zı = 4 (cos 180 ° + i sin 180 °) and Z2 = 4 (cos 145° + i sin…

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

A: According to our company policy we can only answer first question and rest of questions can be…

Q: Prove that (1+i)" +1(1-i)" = 2 cos %3D

A: We will solve the following

Q: (a) Show that z" + z=" = 2 cosn0. (b) Hence show that 32 cos5 0 = A cos 50 + B cos 30 + C cos 0…

A: Given:- Let z= cosθ+ i sinθ

Q: tan z, + tanz, = T-(tan z;) +(tan 72) tan(z, +2) ) Prove that: cos(z + w) = cosz cosw + sinz sin w

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Q: Prove that V1+ cos x +VI-cos x X +÷,0<x<. 4 TT - 1+ cos x -V1- cos x 2 2

A: PS: There seems to be a cut in the image. If this is the question, pls find below its solution.

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: Let z1 = 4 (cos 7π/12+ i sin 7π/12) and z2 = 2 (cos 5π/12+ i sin 5π/12) Find z1z2. Find z1/z2

A: Recall:  z1z2=r1r2cos(A+B)+isin(A+B) z1z2=r1r2cos(A-B)+isin(A-B) We have, r1=4, r2=2, A=7π12,…

Q: Determine whether u and v are orthogonal, parallel, or neither. u = (cos(8), sin(0), 1) v = (sin(0),…

A: If u and v are orthogonal then u•v = 0  if u and v are parallel then u • v = | u | •| v |

Q: Prove that: 2n 1 cos=0 + i sin 0) + 10 2cosne - %3D eino

A: If  is an exponential form of a complex number where x and y are real numbers then its polar form…

Q: Find the points of intersection between r2 =2 cos20 and r=- %3D 1- cose

A: Find the points of intersection between r2=2cos2θ and r=21-cosθ. for the intersection points both…

Q: Prove that: 1. Z, Z2 =r,r2[cos(8, + 02) + i sin(8, + 02)] 2.4=(cos(0, -62) + i sin(0, – 0,)] %3D 3.…

A: Here we use exponential form to represent complex Numbers.

Q: Let z1 = 2(cos() + i sin 12 and z2 = 6(cos() + i sin). Find z1z2 and (Enter your answers in a + bi…

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Q: 1 Prove that sec 03D Cos 0

A: Let P(x,y) be a point on a circle of radius r associated with the angle θ.

Q: Find the product of z1 = 2 (cos 213° + i sin 213 °) and Z2 = 4 (cos 33 ° + i sin 33 °) %3D

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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: 10. Prove that (지12 sin 8 cos 1 (cos a cosec 0) de 1/2-a (1-cos a )

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Q: 8. Show that cos a – cos y| < |x – y| for all æ and y.

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Q: 1- 4cos*x = cos3x + 3cosx 2- Sin(In(i-i)) = 1 3- ((1+i)(1– i))' = (1+ i)'(1 – i)'

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Find all the roots of z²= - /3-j A z=2 (cos +j sin-) B z=/2 (cos (- =+ 2kn) +j sin ( 12 + 2kn)), 12…

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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: Prove that : 1 a) Sec'x= Cos!

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Q: (b) Show that U (x, y) = eu cos v, V(x, y) = eusin v are conjugate harmonic of each other if f = u +…

A: It is given that,  f = u+iv  is analytic. We have to show that,  U(x, y) = eu cos v  and  V(x, y) =…

Q: Find all the roots, of 2=+1+V3j A =D/2 (cos +j sin-) - B z=V2 (cos (-+ka) +j sin ( + ka)), k=-1,0…

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Q: Prove that there exists a value of x so that -1 COS T = cosx %3D ned to

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Q: 5. Determine if u and v are orthogonal u = (cos, sin 0,-1); v = (sin 0.- cos 0,0)

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Q: cos z = cos x cosh y– jsin x sinh y

A: Complex numbers

Q: Determine whether u and v are orthogonal, parallel, or ne u = (cos(8), sin(0), 3) v = (sin(0),…

A: We will check if they are parallel or orthogonal.

Q: Compute cos æ dæ. cos x +C 3 cos a sin x +C sin r cos a + cos x +C sin a +C 3 sin r + sin I 3 sin a…

A: Given, ∫cos3x dx We can write it as by using the trigonometric identity as, cos2x=1-sin2x∫cos3x…

Q: Prove that X + 8 2 TT sin 8 2 1 sin x X sin?

A: We use Trigonometry identities , double angle formulas to prove this Question

Q: Can there exist two solutions y1, 42: [-10, 10] R to the ODE y = V1+ sin (ry) %3D with yi(-10)…

A: The given differential equation is  y'=1+sin2xy On simplifying this, y'2=1+sin2xy. Hence the…

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

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Q: 10. Prove that 7/2 TC sin 0 cos (cos a cosec 0) de /2-a (1-cos a). COS

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Q: Let = r, (cos 0, + isin 0, ) and r, (cos 0, + i sin 0, ). Z, = X1 + iy, 1 Z 2 X2 + iy, Then show…

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Q: Find the points of intersection between r 2 = 2 cos20 and r =- %3D 1- cose

A: We have to find the intersection between r2=2cos2θ and r=21-cosθ.

Q: 4. Express in the form a + jb, (i) 2(cos 30° +j sin 30°) (ii) 5(cos 57° -j sin 57°)

A: To write in the form of a+jb

Q: Prove that ²-1 (²) (0) (sin (+,-) + do r² sin(0)

A: Given : ∇2=∂2∂x2+∂2∂y2+∂2∂z2 To Prove : ∇2=1r2∂∂rr2∂∂r+1r2sinθ∂∂θsinθ∂∂θ+1r2 sin2θ∂2∂ϕ2

Q: 0.3- Prove that: n 2n 1 (cos =0 +i sin8) + 10 2cosne - 5 eine

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Q: Prove that there exists a value of r so that COS I -1 COS %3D

A: Givencosx=cos-1x

Q: 2 Find the points of intersection between r=2 cos20 and r = 1- cose

A: given r2=2 cos 2θ and r=21-cosθwe need to find the intersection between two polar curves

Q: Prove that : A) sin(h3x) =3sin(hx)+4sin(h³ n(n°x) B) 13 cos(hx) 3 C) 2tan(hx) tan(h2x) 1+ tan(h²x)

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Q: 2 Verify the identity 4cos x (20 points) 2 2 2 4sin x cos²x + 4sin²x − 4sin³x = 4cos²x.

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Q: Prove that: tan-¹ z = In( 2 Note: tanz = And sin z = COS Z = ‚i+z. sin z COS Z eiz-e-iz 2i eiz te-iz…

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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: Prove the identities: (x sin 0 – y cos 0)² + (x cos 0 + y sin 0)² = x² + y²

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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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Q: 2. Prove that cos30 = (4cos'e - 3cos0) and sin30 = (3sine – 4sin e) using De Moivre's Theorem. %3D…

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