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 3x cos 4x dx C Show that sin mx cos nx dx = 0 for positive integers m and n: i using the primitive, ii using symmetry arguments.

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: 5. (a) Expand (1+x) and hence find the value of (1.02) correct to 4 decimal place 1+sin 0 m-1 1-sin…

A: First expand 1+x4 then find the value of 1.044 Using the identity, a+b4=a4+4a3b+6a2b2+4ab3+b4…

Q: Q3:- If x= sec 0- cose and y = sec" 8- cos" 0 Prove or disprove that (x2 + 4) ) = n2G2+ 4) %3D

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Q: If x = sec e - cos e and y = sec" 8- cos"0 (x² + 4) (* - n°(y² + 4) = n²(v² + 4) Prove or disprove…

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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: [o,1] Find the sne exponsion of fla)=x+2 in in Sine Sinlntix), %3D 00 4-(-1)^ %3D •Sin (nTx) (c) Sin…

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Q: 2. : (a) For n ≥ 1, show that 1+z+2²+...+ 2¹: (b) Hence, show that 1+ cos 0 + cos 20 + 12+1 2 #1 7…

A: Solution:

Q: To show that rπ/2 sin" x cos" x dx: (m - 1) (m- 3) ... × (n – 1) (n - 3)... (m + n) (m + n - 2) (m +…

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Q: 3. Find g'(x) if g(x) = sin³(4x) %3D A. cos (4x) B. 3 sin?(4x) cos(4x) c. 12 sin (4x) cos(4x) D. 4…

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Q: 1. (a) Use integration by parts to verify the reduction formula cos" x sin x n Scos" xdr = -1 [cos"…

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Q: 14. If we let u = 4x – 1 for the (2tan(4x-1)-3 cos“(4x=1) dx. what will be the cos(4x-1) resulting…

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Q: Let z = 8(cos 50° + i sin 50°) and w=2(cos 30° + i sin 30°). Express in the form r(cos+ i sin 0):…

A: Here we can use the following formula to find the results eiθ=cos θ+i sinθande-iθ=cos θ-i sinθ

Q: 15 and x terminates in guadrant III. 17 Find sin2x, cos 2x, and tan 2x if sin x= - sin 2x cos 2x %3D…

A: The solution is given in the next step.

Q: 10. Prove that T/2 75/2. sin e cos (cos a cosec 0) de 10 T 2 (1-cos a).

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Q: cos(3rk) In k 5k k 1

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

A: The given problem is to prove the following complex identity. (cos n/5 θ +i sin 2n/10 θ)+1/einθ =…

Q: Assuming In = [ cos" xdx. and n > 2: 1 п -1 n-1 r sin x + -In-2; CoS' In n %3D n

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Q: Using the Tabular method, not getting the same answer as my mathlab

A: To evaluate

Q: 1) VERIFY THE POWER REDUCING FORMULA, cos"x : cos 2x+1 HOLDS TRUE FOR 2 x = - AND x =

A: To verify:   cos2x=cos2x+12, for x=π3 and x=π2   NOTE: As per our answering guidelines, we can only…

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A: Wallis's formula state that 1. if  n is odd and n≥3,then ∫0π2cosnxdx=23.45.67 . . . n-1n Therefore,…

Q: 1 (27 x sin nxdx 27 bE f(x)sin nxdx -sin nx -1. 1-2n cos 2nn COs nx %3D %3D The requirod f 2)

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Q: 4. a. Show that for 0 Va² + sin² a. b. Generalize the result in part (a).

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Q: 11.1.5 Show that (a) cosx = Jo(x) +2(-1)" J2n (x), n=1 (b) sinx =2)(-1)" J2n+1(x). n=0

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Q: 4. Let z1 = 24(cos(75°) + i•sin(75°)) and z2 = 2(cos(15°) + i • sin(15°) ). Compute the %3D %3D…

A: SOLUTION-

Q: Determine c1 and c2 so that y(x) = c1 sin 2x + c2 cos 2x + 1 will satisfy the conditions y (π/8) = 0…

A: Given,y(x) = C1 sin2x + C2 cos2x + 1 — (1)Here we find the value of C1 and C2 such that the equation…

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

A: (a)consider the given complex number:

Q: 18.) Suppose x = R cos(0) sin($), y = R sin(0) sin($) and z = x² + y? + z? = R²Y R cos(). Verify the…

A: Given

Q: (-1)"x2" Σ (2n)! 1. For all x E R, cos x = n=0 +00 (-16)"(4n+1) (2n)! c. Use the result in item 1(b)…

A: We have to prove the sum of series.

Q: 7/2 79. Prove that 0< sin x dx < Ja/4

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Q: 1 sketch The graphs O v= 1 between B=o-360 Ov =1+ Sin8 %3D 2 v 2(1- Cose) 3. 5r=I+ Cose

A: We are only allow to solve first three subpart of questions if you need rest of answere then please…

Q: 28. Evaluate f 5 sin 3x dx 2+cos 3x a. 2 In 15 sin 3x| + C b.- In 12+cos 3x|+C c. In 15 sin 3x| + C…

A: Evaluate integral

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

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Q: 1. Evaluate the integral 7/2 cos xesinx *dx

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Q: 1-Find the Cauchy principle value (showing details): J_co (7*+5x²+4)' cos(4x) dx 8.

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Q: . (a) Write cos 3x + i sin 3x = e3ix = (cos x + i sin x)³ by Euler's formula, expand, and equate…

A:

Q: Given: z = sin 3x cos 4y Find: z ду2 -3 cos 3x sin 4y -9 sin 3x cos 4y -12 cos 3x sin 4y -16 sin 3x…

A: Differentiate the given function partially w.r.t y keeping x as constant   ∂z∂y=sin 3x×4-sin 4y…

Q: 1. Express as a sum or difference a.) sin 7t sin t b.) cos 6u cos (-4u) c.) cos 2t sin 6t d.) 2…

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Q: 3. Show f cos2n Ode 277 sin-n Ꮎ ᏧᎾ = 27 22n 11.1 n = 1, 2, 3,...

A: Using Cauchy integral formula.

Q: r(cos0+isin 0) with 0 (exact) in (0°,360°): а. z = 12-12i Z, =-3-3i z, =-4V3+ 4i b. с. Z3 = %3D

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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: Fand ar a of functrons COSX n

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Q: 7.32. Prove that for 05sr, -( . ) cos 2 cos (a) a(e -a) = sin 3r sin ba (6) (r-) - 7.33. Use Problem…

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

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Q: 7π/4 Evaluate the integral (sin x+ cos x) dx using the fundamental theorem of calculus. Discuss 2+…

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Q: Express Q(x) = cos4 (x )+ 12 as the composition of three func-tions; that is, identify f, g, and h…

A: Q(x) = cos4 (x )+ 12 We need to express the given function as a composition of three functions  that…

Q: 2n 1 (cos=0 0 +i sin e) + 10 2cosno eine

A: The general form of a complex number is given by z=a+ib, where a,b are two real numbers and i=-1.…

Q: Prove that : n/2 1 dx (a'sin'x + b’cos'x 4a°b³

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Q: lim (1 - cos(3x))sin(7x) of (c) 이이 O 0° O 100 O o. c0 O co - c0 O not indeterminate 8 %

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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-