13. Write sin(tan-)) ; x > 0, as an algebraic expression of r. 49 + x² iv) 7 ii) 49 + x² iii) 49 + x² v) not listed

Q: (a) Apply the Comparison Theorem (Theorem 5 in Section 5.2) to the inequality sin x 0) to prove…

A: Given: Use the inequality  sinx≤x   (valid for all x≥0)To prove:a) 1-x22≤cosx≤1b) x-x36≤sinx≤xc)…

Q: a) By expanding sin(2x+x) using double angle formulae, show that and sin 3x = 3 sin xr-4sin x.…

A:

Q: 41. Show that if u = tanh(x/2), then 2u 1+u? 1- u?' 2du 1- u? cosh x = sinh x = 1- u? dx = Hint: For…

A: To show the following

Q: b. Prove that 2cot (20) = cot(0) - tan(0)

A: To prove the identity.

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: If y=tanx, Prove that Y™ 2 (Y²+1)(3+1)

A: We will find the required.

Q: dr 4+x² S Choices: tan ¹(x + 2) + C 1 In x² + 4 +C In x + 2+ C 1 tan 2 1 ¹ 1/2 + +C

A:

Q: Prove that L (sin² x 2/3 ; p} = 1/ 108 р x (1²,5 + 4)

A:

Q: 11. Use the substitution u = x/3 to prove dx tan 9+x2

A: Consider the integral

Q: 6. By using u = tan 0, evaluate | tan 0 sec*e de .

A:

Q: a) Show that: tan z, + tan z, tan(z, + 2) = 1- (tan z) +(tan z)

A: To prove the formula:

Q: 12 Prove that sin 80 cos 0-sin 60 cos 30 a. = tan 20 cos 20 cos 0- sin 30 sin 40

A:

Q: | 6. Use a trigonometric substitution to evaluate x³√x²-4 dx.

A: We will find integral using  trigonometric substitution.

Q: Prove the following equation : Cosh(3 x) = 4 cosh³(x) – 3cosh(x)

A:

Q: Let In = fcos" rdx. Show that: 1 In n n-1 COS x sin x + n-1 -¹ n -In-2, n ≥ 2.

A:

Q: 21. Let h(x) = tan(5x + 6). Then %3D h'(2) is and h"(2) is

A: 21) The given function is: h(x)=tan(5x+6) To find h'(2) and h''(2)

Q: 5. Show that de (tan 8) can be expressed in the form 3tan 8+3sec2e-3. Hence, find tan*e de.

A:

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

A:

Q: - IF z = tan2 -1 y Show that x- ây

A:

Q: Find the first four nonzero terms in a 6y"-(cos x)y'-y = 0; y y(x) = +... (Type an expression that…

A:

Q: 13 A Iz =, arg (z) =tan-(8)- 13 B arg (z) =n-tan-(8)

A: the given complex number is 3-2j2j+1 rationalize…

Q: 57 tan 4 177 Determine the value of x if D, sin(7-x)= csc 3 6. where 0sxST.

A:

Q: 46 a Show that cos x+ =-sinx. = -sinx. d" (cosx) = cosx + drn b Prove that

A:

Q: Let S(x) 9 sin x + 9 cos x "(x) 3= r(-)-

A: A function is given as fx=9sinx+9cosx. The objective is to find the first derivative f'x of the…

Q: 5л Compute the exact value of tan ( + 4 ). 6.

A:

Q: 3 then tan 0: sin e = O 5 3. 3 O 3

A:

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

A:

Q: 5. Evaluate: sec°0 tan’0 de

A: Consider the given integral. I=∫sec9θtan7θdθ Further, simplify.…

Q: 6xsec 3x 2-If y = (tan3x)" prove that = (tan3x)²"| + 2ln(tan3x) %3D %3! tan3x

A:

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: Prove that ? 2tan X tan 2X 2 1-tan x

A:

Q: 7. Suppose x is a number such that cos(x) = sin(x). Then: O a) cos(2x) = cos(x) O b) sin(2x) =…

A:

Q: 4.) Prove: sin 2x a) 2cos? x = tan x b) sin 3x = 3 sin x – 4sin3 x [Hint: sin 3x = sin(2x + x)] -

A:

Q: Prove (05t - sin 4t tos?t

A:

Q: v3 V3 sin 30 = , cos 30° = , tan 30° = COS %3D O sin 30° = , cos 30° = . , tan 30° = %3D %3D V3, cos…

A:

Q: (b) у — 2г tan-"(г) — In(1 + x?)

A:

Q: (b) Can we use the IVT to conclude that f (x) = tan(x) equals 0 for some c in (0, 1)? %3D

A: We can answer first question among which you asked as per the honor policy. please resubmit the…

Q: Q4) Prove that 1- Cos +x) = sin x, 2- (sin x +cos x) sec x see' x

A:

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

A:

Q: 30. 7/3 tan'x sec'x dx

A:

Q: 1 .Prove that : sec q + tan q = 2x or, 4x If sec q = x+ 2x

A:

Q: pose m and n are const (-)+(+)x] + C m-n m+n [-cos(m-nx- -cos(m-n)x - cos(m + n)x] + C 1 m+n…

A:

Q: 3) Prove that b) tan- (*)} = sechx

A: Evaluate the derivative of the given function as follows. ddxtan-1ex=11+ex2ddxex…

Q: Prove that? 2tanX tan 2X %3D 2 1- tan x

A: Given : 2tanx1-tan2x = tan2x

Q: Q3:A| Prove that:- cos x 1) 1+sin x 1+sin x = 2 sec x cos x tan? x 2) tan? x+1 = sin x

A: topic - trigonometric identities

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)

A:

Q: (6) Given that In= tan" x dx, show that, for n 2 2. %3D In- In-2 = %3D n- 1

A: I think given question is slightly wrong. According to me the required proof should be…

Q: a. Show that y = tan u increases on every open interval in its domain. b. If the conclusion in part…

A: (a)  Given,

Q: what is the deivative oF: tan(3x) eX y = In x

A: The given function is: y=2xtan(3x)ex8ln x

Q: (d) Show that there is some x with 0 < x < 2 such that x2 + cos(ax) = 4. %3D

A: Solution-