4. Verify the identity sin(z+y) %3D COB E COS y tan z+ tan y

Q: Determine the value of f(z) = e*cos y+ ie*sin y at z = In 4+ Ti/6

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Q: 7. The value of A. cosx - C O A O B О с O D dx √(1-x is B. sin '(x)+c C. cosx ' +2¢ D. cosx

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Q: S cos“ xdx = sinxcos² x + sinæ + c Select one: O True O False

A:

Q: Find cos (z+2) dx. O-sin (x + 2) +C O sin (x4 + 2) +C O sin (x* + 2) +C sin (2* +2) +C

A: Let's find.

Q: 1. Differentiate the following functions: cos (x)+1) sin-(2x), a) y =

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: If z = +i then Im(coshz) A) –sin1sinh4 B) cos1cosh4 e* +e* C) cos1 2 D) sin1cosh4 E) sin1

A: Topic:- hyperbolic function and complex number

Q: 10. 3 J√cos z sin ³ zdz COS

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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: A certain substitution gives S sin³ x cos" x dx = S(u7 – u5) du. What is n?

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Q: if y = a sin x Prove a +b CosX a cos + b %3D 2 (atb cosx) 11

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Q: du nd az and " if z = 5 tan-'(x)and x = e" + In(v) at tl-

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Q: (x cos + y sin?)y – (vsin? -x cos?)* х — х COS = 0 Ans: xy cos = c (Lucknow, 1997; Kanpur, 1996)

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Q: 9. dt + 을 = COS t + Answer: s = sin t + 을..

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Q: the solving for equa tion ty=o is 1-y= COSX %3D 2-y= Sin x %3D y= tax

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

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Q: Show that 0 π/2 st sin? xdx TT/2 [ 0 cos? rdr 2

A: A detailed solution is given below.

Q: Sketch the following between x = 0 and x = 4 pi      y =  sin x + cos 2 x

A: Given: The function is  y =  sin x + cos 2 x

Q: Suppose m and n are constants and m‡ n. 1. Evaluate sin(mx) cos(nx) dx

A: using trigonometric identity

Q: 0° S (c) Prove that: 2cos 20. cose 2 + 2 tan0. sin0 = %3D cos?e – sin? e cose o AEGW is drawn.

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Q: Show that COS x dx a² – x² sin a 2а where a is some positive number.

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Q: help!

A: Refer to the question, we need to sketch the provided function.

Q: What is the general solution of sin z sin ydx + cos rcosydy 0? %3D

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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: Suppose m and n are constants and m #n. Evaluate sin(mx) cos(nx) dx-

A: To evaluate we first multiply and divide by 2 the given function.  Then use the formula: 2(sin(mx)…

Q: If sin x = siny = a and cosx = cosy = b, then cos(x-y) is Select one: O a. z2 + a² O b. 2b + 2a O C.…

A: Given  sin x = sin y = acos x = cos y = b Use cos (x-y) formula cos(x-y)=cosx cos y + sinx sinysin…

Q: 22. If y = (1- sin?x)i then y" is b. 1 COS X 2° а. C. - cos x d. -sin x sin x HIN HIN

A: The second derivative of the given function is solved in detail below.

Q: Evaluate si (z* - tan(z) cos(z) ) dx 2-1

A: To find the integral: ∫11x2-tanxcosxx3-1dx

Q: 8. Evaluate (a) /sin z cos" z dr (b) / (ln z)* dz. You may use /In z dz = z ln z - z+c.

A: Let's find.

Q: 14. Find dr/de: r= 0 sin 0 + cos 0 а. cos @ b. 0 cos 0 0 sin 0 C. O a O b

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Q: 7 (a² +b°) 1 Prove that : dx = ở (a sin x+ b° cos² x)' 4a’b?

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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: e* (1 + sin x de. nx] Evaluate 1 + cos x

A: I am attaching image so that you understand each and every step.

Q: 2Tt Prove that a" sinax cos bx dx = 0 0.

A:

Q: 4) Differentiate each of the following: a) y = - cos²x c) f(0) = sin(20 — π) b) y = sin(20) - 2…

A: Differentiate the following  a) y=-cos2x b) y=sin2θ-2cos2θ c) fθ=-π2sin2θ-π

Q: If g(x): = cos 2x and f(x) = sin-n, find g(f(x)). %3D

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Q: Cos X 2)

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Q: 1. Given sinx 12 and cos y : 13 both x and y lie in sxs, determine a) cos(x+y) b) sin 2y

A: Given query is to find cos(x+y) and sin2y.

Q: . Evaluate [10x cos(x²) dx . A) sin(x³)+C C) 30 sin(x') +C 3 B) sin(x') +C 10 D) 3 sin(x') +C 10

A: Our Aim is to evaluate the integral given below:-∫10 x2 cos (x3) dx -(i)

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

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Q: 4) Find when y2 – x² + cos(x + y) = 0

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Q: Find y prime for y=(cos(x))^x

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Q: y = C, sin at + C2 cos at, (a not to be eliminated)

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Q: 9. Integrate the following: a) S sin? x cosx dx b) "(1 – cos 2x)/2dx

A: (b) The given integration is: ∫0π1-cos2x32dx To find the value of integration.

Q: A certain substitution gives S sin*x)cos"(x)dx = S (u²-u*)du .what is n?

A: For substitution- Balancing the above equation, we can solve it as follows-

Q: 1) Using the formula: cos(x + y) = cos x cos y – sin x sin y, prove in steps that: cos? x = %3D…

A: Note - Since you have posted multiple questions, I am answering first one for you as per the…

Q: 8. Evaluate (a). (20-0y di (b). / COS I sin(sin x) dx -3

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Q: Differentiate implicitly to find 04 0-3 O 3 3 cos(y+z) 4 cos(y+z) Əz Əy 2 given 4x + sin(3y+z).

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Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.