7. Show that sin E + x) = G+x): = -cosx.

Q: Prove that the equation sinx = 0.3 has at least one solution in the interval (o, ~).

A: Given:sinx=0.3

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Q: Q1/ Show that 3(sinx-cosx)*+6(sinx + cosx)²+4 (sinºx+cos°x) = 13

A: Note: We are entitled to solve only one question at a time. To Show:…

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Q: 3) Prove that c) cosh2x = 1 + 2 sinh? x

A: In the question we have to prove the given function.

Q: Prove that  d/dx[cosx]=-sinx

A: We will use the definition of the derivative  Here f(x)=cos(x) So,

Q: s nip

A: Given z=tanα2

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

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Q: 12 "2 If Y= \n(Sinae), Prove y =y=- ノ

A: Here, the given function is: y=ln(sin x) So, y'=1sin x(cos x)=tanxy''=ddx(y')=ddx(tan x)=sec2x

Q: 3. Prove that the identity is true. sin*t- cost = tan²t-1 cos't

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

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Q: e cosl – i e™ sin 1 + e¯ cos 1 - i e- sin 1 2 e cosl + i e™ sin 1 – e- cos 1 + i e¯I sin 1 -TT 2 e…

A: Simplify the given expression: Coshπ-i

Q: 1- d/dz [cosz] = -sin z

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Q: 5. If y= tan x, prove that y``=2(1+ y² )(3+ y²).

A: Prove the given equation

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Q: (3) Estimate the error involved in using the approximation cos x = 1- for |x| < 0.01

A: To estimate the error involved in using the approximation cosx≈1-x22 for x≤0.01 The Taylor series…

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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: Use linear approximations to estimate the following. 1. sin 9/4

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Q: If f(x,y,z)=In(x²y+sin² 12

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Q: Compute : exp(i sin(e-i))

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Q: If xy-In(3 sinh(3x)) , then x? y'-

A: Given,           xy=ln3sinh3x

Q: (§5.4, Q42) Calculate So™ f(x) dx, where sin x, x T

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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: Evaluate the integral. sin (8t + 16) dt cos (8t + 16) sin (8t + 16) dt = cos (8t + 16)

A: Solved below

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Q: 11.. , Evaluate each definite integral. a) cos² x sin x dx

A: Use substitutions to integrate

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A: Proved Coshx - Sinhx = e-x

Q: 12. Write an éxpression that gives all sölutions to the equation. COS X - SIn x 3= 2.

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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

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

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Q: EX/ If f (x)= + COS2X CoS X is Pcx) Continlacaus at x=? %3D

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Q: Use the addition formulas to deriv TT 31. cos(x - sin x 2

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Q: B ) if y= sinh¬1x then prove that =

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Q: Show that, for real values of x, 3 sin x f(x) 2 + cos x cannot have a value greater than 3 or a…

A: To show that the minimum and maximum value of the function is -√3 and √3

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Q: Show that 1 − cos x = x2/2 with an error less than 0.003 for |x| &lt; 1/2.

A: Given: 1-cos x =x22;    |error|&lt;0.003Formula:  The…

Q: 7. Prove that: ++) + sin; + e) = v3cose (3) (a) COS

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