Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the square root of −1)

Q: Express as a product:cos 3θ - cos 8θ

A: Known fact: cosC-cosD=2sinC+D2 sinC-D2.

Q: If y= In(4 tan (4x) ), then y' tan(4x) = Select one: 04/(1+x² ) O None O 4/(1 + 16 x² ) 01/(1+ x² )…

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Q: Determine and of y = (5r - })(cos z + In z

A: Consider y=5x2+32sin x+ln x. Calculate: dydx=y'. Apply product rule of derivative:…

Q: Show that (cos 0 + i sin 0)" = cos n0 + i sinne for n = 3 and n = 4.

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Q: /2 1lf If In = " x cos" x dx, when n> 1, show that п(п — 1)1-2 — 1 In 12 [Hint: cos" x = cos²…

A: Given: In=∫0π2x cosnx dx

Q: show that cos 6x = cos 2x (4cos2 2x-3)

A: Given,           To prove: cos 6x = cos 2x (4cos2 2x-3)

Q: Prove using the IVT cosx = cos-1x has a solution in (0, 1).

A: cos x = x   rearrange to x - cos x = 0

Q: If n is an odd natural number prove that I =, xsinnx dx = sinx 2.

A: Consider the given: It is given that n is an odd natural number, Take n = 1,Substituting n=1 and…

Q: 5) Integrate 2x cos (x²-5) with respect to x.

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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: Find an expression that is mathematically equivalent to (1 - cos(x) / 2) and in which excessive…

A: We will find the expression mathematically equivalent to the given term as following.

Q: Show that sin mx cos nx dx = 0 for positive integers m and n: i using the primitive, ii using…

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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: Show that (cos 0 + i sin 0)" = cos n0+ i sinne for n = 3 and n = 4.

A: Since you have asked multiple question, as per company guidelines we will solve the first question…

Q: Suppose n is an integer. So Show that for n+0, ſ x cos(nx)dx = { п n is even (a) 2 п is odd n2

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Q: 39. Does not exist; cos e – 2 lim 00- cos e – 2 lim 00+ and

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Q: Prove cos 4x = 8 cos4x − 8 cos2x + 1.

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Q: Show that sin 3x and cos 3x are orthogonal on [−π, π].

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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: 10. Prove that 7/2 sin 8 cos (cos a cosec 0) de =- (1-cos a). CO TE/2-a

A: We will use the Differentiation under Integral sign method, to solve the problem.

Q: cos" xdx :0. 6. If n is a positive odd integer, prove that

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Q: Can you show me a step-by-step proof

A: We need the following two identities to produce the required prove:

Q: Prove the following: sin² 0 – cos² 0 = 1– 2cos² 0 %3D

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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: Use residue to evaluate the integral 3+ 2 cos 0 do 5 + 4 cos 0 0.

A: Given integral is, Let C denote the unit circle |z|=1. So,

Q: 10. Prove that 7/2 sin 0 cos (cos a cosec 0) do -(1-cos a). 2 ||

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Q: Show that sin (2n + 1)z 2n +1 { 7/4 */4 0<まく

A: We will use complex analysis to solve the given infinite sum.

Q: 2c. Show that * (5 - ) 4 sin z - cos 2r + 3 -8 cos for every r.

A: To prove: 4sinx-cos2x+3 = 8cos4π4-x2 We know that cosx-y  = cosxcosy+sinxsiny and cos2x= 1-2sin2x

Q: Sın 7. os 3x-Cos 72- Sen 3x tan 22 Cos4a+1

A: We use Trigonometry identities  Sin(A+B) = sin(A)cos(B)-cos(A)sin(B) cos(2A) = 2cos2(A)-1 sin(2A)=…

Q: Express cos 60 as a polynomial of cos 0: Cos 60

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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: Prove the reduction formula: cos" xdx : 1 -cos"- 1-1 x+ cos"-2 xdx %3D -

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Q: 7. Suppose x is a number such that cos(x) = sin(x). Then: O a) cos(2x) = cos(x) O b) sin(2x) =…

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Q: Prove that Σ n=1 sin ne n - 6.² < sin x X - dx = 1.8519... for any positive integer m and any 0 in…

A: In this question we have to Prove the following given statement. ∑n=1m sin nθn&lt;∫0π…

Q: Show that n. In(cos 2k+2) = - In (2"+ sin 2n+2). k=0 Hint: sin 20 = 2 sin 0 cos 0.

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Q: Show that 2 cos(x) = 3x has exactly one real solution.

A: As we know that if the function is strictly increasing, f'(x)&gt;0 or strictly decreasing,…

Q: Let m be the number of solutions of sin (2x) + cos (2x) + cos x + 1 = 0 in 0<x< x< 1/1/2 and 7 7π n…

A: The roots of the equation: f(x)=0 are the values of x, where the graph of the function: y=f(x)…

Q: - Find all solutions for 4 cos x+V3 = 2 cos x if 0<x<2n. %3D

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Q: Prove thet (1+ cose -isine)" +(l+ Cos e+isine)" = 2""cos (em). %3D Co."(을) COS

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Q: Use the fact that (-1)"x²n cos(x) = L(2n)! n=0 for all x eR to evaluate cos(x²) dæ, correct to…

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Q: prove it 5. 2 sin (3x) = 1– cos(6x)

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Q: -) {in(6n) 2 n=2 =) an = In(6n³ – 4) – ) {cos(nt)}0 n=0

A: e) an=ln6n3-4-ln3n3+5n2-7n+8 determine if the series is convergent or divergent? if convergent find…

Q: Use residue to evaluate the integral 2л 3 + 2 cos 0 do 5 + 4 cos 0

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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: find y prime and y double primey=cos(x^2)

A: To apply chain rule and product rule to find the first and second derivatives of a composite…

Q: 40. Show that cos(30) = 4 cos e - 3 cos e for all 0. [Hint: cos(30) = cos(20 + 0).]

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Q: Prove cos 4x = 8 cos4 x - 8 cos2 x +1.

A: In trigonometric ratio,cos2θ=2cos2θ-1

Q: 10. Prove that 7/2 sin 8 cos (cos a cosec 0) de re/2-a (1–cos a). 2

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Q: (d) Show that there is some x with 0 < x < 2 such that x2 + cos(ax) = 4. %3D

A: Solution-