these questions have the answers but i want step by step how to solve them

1. Prove the following identities: (a) secr - cost = tan r sinc (b) sin(r + y) sin(x - y) = sin x - siny 2. If cost = -; 3 7 0 1. Answer: * > -5 10. Without using a calculator, evaluate: tan ( sin- Answer: 21/2 11. Evaluate the following limits, if they exist. (Do not use L'Hospital rule) (a) lim Vh' + 15 - Vh + 15 Answer: 0 Vh +3 -2 (b) lim Vh2 + 15 - Vh + 15 h - 1 Answer: 1/8 2 - 3x + 2 (c) lim -2 212 - 30: - 2 Answer: 1/5 (h + 4)2 - 16 (d) lim Answer: 8 h 24 - 3r' + 10 (e) lim r+1 x2 + 2x+1 Answer: 2 (f) lim x - 1 Answer: LHL= -1/2 and RHL=1/2, so the limit DNE (g) lim(* +7 )(#') Answer: 0 using the squeeze theorem (h) lim sin (In x) In r Answer: 0 using the squeeze theorem () lim V4x2 - +1 -2x 3c + 2 Answer: -4/3 (j) lim 5x2 - 3x + 1 9:12 + 2 Answer: 5/9 (k) lim v16x- + 7: -4x Answer:12. Find the horizontal asymptote(s) and vertical asymptote(s) of the following curves. - 4r + 4 (A)= 2 + 5x - 14 Answer: HA: y =1, VA: x = -7 (b) y = 2-20 - 36-2x Answer: HA: y = 1, y = -}; VA: x = } In(1.5) 13. Find the value(s) of c that makes the function continuous on R. f(x) = if x >1 Answer: c = 4, -2 14. Show that the equation cosa ty-3 =e- has a real root. 15. Use the definition of derivative to find the derivative of f(x) = a + 5r. Answer: f'(x) = 2x + 5 7 16. Use the definition of derivative to find the derivative of f (x) = Var + 4 35 Answer: f (x) =- 2(5r + 4)3/217. Find y'. You do not have to simplify your answer. 2x (a) y = sin(cot) + (x + 3)1/3. Answer: y' = cos(cote)(-csc-c) + 3(2:2 + 3)2/3 2" + cosa + 3 cota2 (b) 1 = 1 + 9r (23 In(2)6r' - sinx - 6resc'(x?)) (1 + 9r) - (27 + cosa + 3 coto ) (9) Answer: y' = (1 + 9:r)2 (g) y = e * sin(2 - 2r) - ex: Answer: y' = 5ed sin(x3 - 2x) + (3x3 - 2)ex cos(a - 2.x) -esecc tan Ary vry - y (d) vay = 1+my Answer: 1 = - 2x2 /xy (c) ysine = xsiny' Answer: y' = 2ry cosa2 - sin y' 2.ry cosy? - sina 18. Find an equation of the tangent line to the curve a try-y-1=0 at the point 3 (1, 1). Answer: y = 50 - 19. Find the point(s) on the curvey = 2x3+3x-12x+1 where the tangent is horizontal. Answer: (1, -6), (-2, 21) 20. Find the values of a in [0, 2x] where the tangent line to the curve y = 2sing - 5r is perpendicular to the line r - 4y = 7 Answer: ( = , or c = 3