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

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: 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: H.co/prove That;- CoS z

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Q: Prove that the inequality|cos(x)|≥1−sin2(x) holds true for allx∈R

A: Given inequality is

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: 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: 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 all values of x between 0 and 2pie that satisfy the quation.   cos x = -1/2

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Q: S Show that L(cos² 3t – sin² 3t)= 82+36

A: Using Laplace transform,

Q: Show thatI cos ve (-1/4s)

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

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Q: Given f (x) = |x| for x E [–n, 1] Show that 4 cos(2n – 1)x |x| = 2 E - - - (2п — 1)2 n=1

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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: EX/Show that cos"x (1+ cos2x)

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

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Q: - IF z = tan2 -1 y Show that x- ây

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Q: Find дх (дхду) 6,e at the point(7,), where g(x,y) = cos(y) [sen(x) + xy].

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Q: If f(x,y,z)=In(x²y+sin² 12

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Q: a) By using convolution theorem, show that 1 (sin 3t – 3t cos 3t). 54 1 L-1 (s² + 9)2

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

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Q: Show that (cos x + isin x)n= cos nx + isin nx whenever n is a positive integer. (Here i is the…

A: We can prove given result by mathematical induction: Define a statement p(n).  Show that p(n) is…

Q: 1. Prove the following identity (4) cos(2x)cot(2x)=2-cos(x)(2x)+sin(x)(2x)

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Q: Show that: cos z=0 if and only if z (/2) + kT.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Prove that the inequality |cos(r)| 2 1- sin (x) holds true for all r ER.

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Q: Given f(x) = |x| for x € [-n,1]| Show that cos(2n – 1)x |x| 2 \xl = -- - (2n – 1)2 n=1 2.

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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: 8. 6x° cos (x³) dx

A: ∫ x8cosx3dx

Q: Evaluate the integral: 3 X sin ³ COS 3 3 X col 3

A: Solved below.

Q: Show that: cos z= 0 if and only if z (/2)+ kT.

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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: What is the multiplicity of the zero p = f(x) = 2 – 2 cos x? of Select one: 1 2 3 4

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Q: 4 Giren tan A =– where 0<AS Find 2sin 2A.

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

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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: 1. By considering (1+ expi®)", prove that "C, cos r0 = 2" cos"(0/2) cos(n /2) (ii) ►"C, sin r® = 2"…

A: First, expand the given complex function:

Q: Prove that 55 sin x² dx - 5 cos x² dx = 1/2 √

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Q: (b) Prove that 2 -3/2 Js12 (x)dx = sin xy + C. 'COS X x2 x3

A: We have to prove ∫x-32J52(x)dx=2πcosxx2-sinxx3+C

Q: Q/ IF y = Sim ܠܐ 10²² (4) y = 3/² sin(x) cos(x) Prove that

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

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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: 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: Q6:- Prove that: y" = 0 if (cos y3csc COS

A: To prove : y'' =0

Q: Given f (x) = |x| for x € [-,1 Show that cos(2n – 1)x (2n – 1)2 4 |x\ = ; 2 n=1

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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-