Fix a real number r # 1. For all integers n > 0, 1+r' +r? + •..+ r" = r – 1 pll+1 – 1

Q: Prove that 1-x ', V 2 1+x cos'x = 2 sin 2cos 2 COS 1 -1

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Q: 2. Prove: cot2x - csc²x = -1 %3D

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Q: 5. Prove that So*(y – 0)dydr =

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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: 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…

A: To prove given statement we need to first convert the polar co ordinate to Euler form then use…

Q: Which numbers satisfy lal = a? Which satisfy lal = -a? What about I-al= a?

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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: 2 Use the figure to identify all angles between 0 and 2n satisfying the condition csc (0) = -- V3…

A: Given that, There is a figure in which angles are pointed and cosec θ=-23 We have to find the value…

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: (a) Verify that tanh P-tanh Q 1-tanh P tanh ọ tanh( P – Q)

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Q: 2. Show that c(x, t) = solves 1 ac(x, t) 2 ax ac(x, 1) ar

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Q: find maclurin formula in the fourth degree for f(x)=secx

A: To write the maclurin formula in the fourth degree

Q: 1. Use the Intermediate Value Theorem to prove that the equation 3x - 4x? = 3 is solvable on the…

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Q: Show that |cos x - 1| <=|x| for all x-values. (Hint: Consider ƒ(t) = cos t on [0, x4].)

A: Mean value theorem states that

Q: 7. Use the Intermediate Value Theorem to find an interval of length 1/2 in which the equation In r =…

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Q: Use Fermat's Theorem in your proof that Vn E Z>², 2 is irrational. I

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

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Q: Prove that for all whole numbers n ≥ 1,

A: Explanation: Given that,  n≥1dndxnsinx=sinx+nπ2   Formula Used: 1. ddxsinx=cosx 2. ddxcosx=-sinx

Q: Find all 0 in the interval 0 <0 < 2T satisfying tan(0) = - 0. Give all answers as exact values…

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Q: Solve 2cos (x)- sin(x)=2 exactly for all x in the interval [0,27).

A: 2cos2x-sinx=2⇒21-sin2x-sinx=2…

Q: 6-prove that the Sum of oots of th equation 2X+3X+5=0is=3 the multiple is I

A: Sum and product of roots of qudratic equation

Q: 3. Give upper and lower estimates (in terms of n alone) for sin nb COs na + valid for all a, b; show…

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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: Prove that d (sec"x)= 1.

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Q: 3. Prove the pythagorean Theorem

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Q: 3. Prove that: tan-1z = Z 2 in () i-z.

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Q: verify the set {∅n = cos(nx) (∞ n=0) is orthogonal on the interval [−pi, pi]

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Q: If x,y,z are real numbers satisfying the equation 25 (9x* +y° )+9z²-15(5xy+yz+ 3zx )=0 then prove…

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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 : <(tan"e) - Ve +1 2e 2 +. Ve? +1 tan

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Q: Prove that z'e'-z = 1 has exactly two roots inside the circle |z] = 1.

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Q: Prove that the equation sinx = 0.3 has at least one solution in the interval(o, π/2).

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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: Prove that the equation * + + = 1 has no solutions in positive integers x , Y, Z.

A: x/y + y/z + z/x = 1 if x , y , z are positive integers then case 1 :  If all three numbers are…

Q: Find the root (3 – 3i)2 - + V18[cos(r/8) + isin(x/8)] O +V18[cos(x/8) – isin(t/8)] +3 V2[cos(1/4) –…

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Q: -Prove that: y" = 1 if y=csc(tan")

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Q: Q3. Prove that tanhz =: 1-22

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: find maclurin formula in the fourth degree for f(x)=tanx

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Q: Let z=x+iy be a complex number such that coshz#0. Then Im(tanhz) is equal to

A: None of these is correct.

Q: Let C be the shortest path that connects 7 to ni and I = f. (coshz + 1)dz. Then --sinh π+ π(1- )…

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Q: 24. Find the Fowrier r fas - -IN X

A: Hv

Q: What is the midpoint between z1 = 9i and z2 = 4? (2, 4.5) (-4,-9) (-2,-4.5) (4, 9)

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Q: Use Newton’s method to find thezeros of ƒ(x) = 4x4 - 4x2 using the given starting values. x0 = -2…

A: Given:                   f(x)=4x4-4x2   Using the starting value :…

Q: positive and r ility of obtaini=

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Q: Prove that (a) Pn (cos 0) = 2F1(-n,n+ 1; 1; sin²0/2).

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Q: tan 4X Prove that 4t(1-ビ) if 1-6t +tf t=tan

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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: Show that f(2) = (2-1) cos) singue has an essential z + 2 at 2=-2

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Q: Determine the ratio and interval of con (-1)"x" in=0 n6" O R.O.C=6, 1.0.C: x E [-6, 6) R.O.C=1,…

A: Here we have ∑n=0∞(-1)nxnn6n=∑n=0∞an ⇒an=(-1)nxnn6n

Q: riemann from 0 to 2 of 2x+1

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