Show that p² - a² L[x cos ax] = (p² + a²)? ' and use this result to find 1 |(p² +a²)° _

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: 2. Find an interval where f,(x) = sin2 x and f2(x) = cos² x are linearly independent. %3D

A: W(π/4) = -sin(π/2) = -1 W(π/6) = -sin(π/3) = - √3/2 Since, Wronskian is not zero for a single value…

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

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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: s nip

A: Given z=tanα2

Q: Determine the order of the zero at the origin of f(x) = ² (cos(z) - 1 + sin² (2))

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Q: find the volume of the solid

A: Find volume of solid.Sketch the given function.

Q: Q4: Prove that: (1- sin x)(1 + cscx) = cos x* cot x %3D

A: To Prove1−sinx1+cscx=cosx×cotx

Q: リ=!-Cos2x 1+(os 2x prove y = 2 tanx .secx

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Q: 26. i) differentiate, with respect to x a) cos (2Vx) In(2x) b) 27. Differentiate y with respect to x…

A: See the attachment

Q: 4. Show that f(x) = sin(x² + 3x) is equal to g(x) = -x cos(x) at least once on %3D Hint: Think about…

A: Given that, f(x) =sin(x3+3x),  g(x) =-xcos(x) Given an interval as -π,-π6 To show that f(x) and g(x)…

Q: 82. Show that cos at cos bt %3D In

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Q: 18 a Show that / (1 + sin 2zx)dx = n, for all positive integers n. b Sketch y = 1 + sin 27x, and…

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Q: if f(2) = zsin(z) -1 defined on the interval (2. 3] then es using False-Position method equals [x in…

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Q: (1) If y = sin(tan-x), prove that %3D (1+ x²)?y" + 2x(1 + x²)y' +y = 0.

A: Since you have asked multiple questions in single request.so we will answering only first question.

Q: Prove using the IVT. tan-1x = cos-1x has a solution.

A: Given :                      tan-1 x = cos-1 xSolve :                     tan-1 x = cos-1 x…

Q: Use the trapezoidal rule withn= 4 to approximate %3D cos(x) COS

A: Given: n=4 and ∫15cos(x)xdxwe know that trapezoidal rule for approximating ∫abf(x)dxis given…

Q: E prove that - Cotx + C %3D 2 |- Cos?x +Sinx

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Q: Use the Intermediate Value Theorem to show that the equation () 2021 E) - cos(t) = 0 as at least one…

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Q: - IF z = tan2 -1 y Show that x- ây

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Q: E prove that = - Cotx + C %3D 1- Cosx +Sin'x

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Q: (a) Show that cos(x²) > cos x for 0 ;. 1

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Q: Q1: show that 1) d/dz [cosz] = -sin z, * 2) 1+tan2 z = sec2 z

A: 1) ddz(cos z)=-sinz2) 1+tan2z=sec2z Results:ddxf(x)=limh→0f(x+h)-f(x)hlimx→0sinxx=1limx→01-cosxx=0

Q: 9 If f(0) = cos 0, cos 02 cos 03 ..... cos 0n %3D Show, {tan 0, + tan 02 + tan 03 + ......... + tan…

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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: Prove that v2 cos(20+45°) = cos e-2sin Ocos 0-sin 0, where 0 is measured in degrees.

A: *formula:- cos(A+B) = cosA*cosB - sinA*sinB. sin2A = 2sinA*cosA. cos2A = cos^2(A) - sin^2(A).

Q: cos 2x +1 sm 2x Prove that Please show proper steps sin 2x 1- cos 2x

A: We know the formula (1) 1+cos(2x)=2cos2x (2) 1-cos(2x)=2sin2x (3) sin(2x)=2sinxcosx

Q: 4. A force of F(x) = x - cos(3x) + 2,x is in meters, acts on an object. Identify the work required…

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Q: Show that the equation 2x + cos(x) = 1 has exactly one root. (IVT and Rolle's Theorem)

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Q: Prove that: cos(nx) - cos(na) cos(x) – cos(a) sin(na) sin(a) dx = n neN, a ER - {0}

A: To solve this question take different values of n and calculate LHS and RHS separately.     if…

Q: .If z = e-(sin x + cos y), show that a²z a2z az ax2 ду? at

A: Given - z = e^-t( sinx + cosy)

Q: 7. B- Show that: (1) lime-0 tan 8 (2) tan 0 sec20 =D1 %3D do

A: Solve

Q: 2- y = prove that (sin x) = cos x %3D dx

A: Topic:- derivatives

Q: .2 1 2x -1 Prove that : sin .2 dx = 72 1+x 1+x?

A: Solution of given question is given step by step

Q: x sin 2x sin COSX 8. Prove that: %3D 2x-7

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Q: (2x) Sinx 2. Prove that (1) Jy₂ (x)=√√(x) =_ (11) J-y₂(x) = √(3/1²) Cosx

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Q: The suitable substitution to evaluate the integral cos°(2x) dx is 5(2x) 3 csc°(2x) u = sin(2x) u =…

A: First, simplify the integrand and use the substitution method

Q: Integr S√₂-cod -1 ation fuctions. 1-cos²x) dx Not of the above B 1 O

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Q: Find (algebraically) the absolute maximum and minimum values of f(x) = cos² x+ sin 2x on [0, 5), if…

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Q: find do diy y = (cos(x))*os)

A: Given y=(cos(x))^(cosx) Now differentiate y with respect to x

Q: show that1) d/dz [cosz] = -sin z , 2) 1+tan2 z = sec2 z

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Q: Can the Intermediate Value Theorem be used to prove that the equation 9 2(r + 9) sin*(x) – 4 cos"(x)…

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Q: a3z cos(x – 4y) show that a3z 6. For z = дудхду

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Q: Evaluate the integral. sin (6t + 17) -dt CoS 2 (6t + 17) sin (6t + 17) dt%3D cos 2 (6t + 17)

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Q: The approximation of 1 = f*cos(x³ - dx using composite Simpson's rule with n= 3 is:

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Q: 3 Differentiate: y = In cosh° (cos 2x)

A: As we know that: The chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1 From the given problem:…

Q: H.co/ prove That;. dz 1-2 Cosh2=

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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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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.