Question

Q5 (20 Mark ) Answer the following 1+cOS X a) lim - tan2 x b) Show that dy {tun-'(see x + lan x)} = { cot-'(esex + cotx)] c) Evaluate if ytan't. and x In(t D) att 1. dx العرب عبر العال مع Google مطلقا تو انشاء هذا النمودج داخل Southern Technical University. الأيام عن اسامه استخدام

Answer 1

OCCO o (a) lim Itcosac at i0 Los a--/ = IT COU OC = 0 XTI tan 2x at tan al =0 -) tan2x = 0 So, this is a form · So, we will apply L'Hospital method a lim d (l+cov) = lim - sina = lim - sinh 2 finx XL XJT da 2 tanx secc d (tan²x) cos²x asi -x-1= - lim I cost I cos 3 a final answer x-TI 2 3-) TT 2 TT COSIL - // cons

2)] - Bylincarity LIS = t (i+sin32 + cosa cosa 2 LH5= d s tan-' (tanac tsecac= d tani' (tans tsecu)x d (tanat seca) dre d (tanxtveca) dsc et tanat scox it L By chain Rule d tan.'(t) xld (tanı) + d Cueca) dt da asi Now altan"(t) = 4 d Itanlı) Esecx d (1x) -tansevera dt Ht² dri da Standard formulae ( sec?xtsaxtanz) (secx ttanx) 2+1 (Itsinx) ㅗ sina = Otsina) ㅗ - LHS costa cos²x Cus 34 (Itsina)? Hos²x Tot sinz ) +1 COS tod22c It sina tribal 1 tsin?xt2sinac tlos? 2 (tsinx) RHS = Scot(cosecx twta)} = d cot" (corecz tcotx) vd (coseextcot x) dx d (cosecetoota ut cosecx tootx ot Ed cot-'(t) x (d cosecx + d cota (d casecx + a cotx] -By linearity dt dk as Now d cot-'(t) = d cosecx = -lota cosecx, d cotx = E-coseca at Itt? dx ds Standard formulac RHS = X C-cota coseca - Cosec? x) H(cosecx tcotx)? It cosa CosX X I t I Itcosa sina sing sinza sin 2x sin?actitros?xt 200sx t cost sin ct{1+cos uc)? sinx sinx sinh dru 2 It 는 Flora 2 (Host) = 1 2 (i.e. 41 = R145= { Hence browen)

ans dy da dy - d tan-'(t) 1 (a tan't = I IT dt at Itt2 standard Itt? asi at at d en ct 241) at x d (t?+1)_ Chain Rule - den(t?!D) x d (+²40) d(+²+) dt (K:11) IX at (dlink to = Itt? Ittz denk X dit?41) K dk dt dk K standard Formula = at 1:1 = 1/2 ㅗ at Itt? I xat Itk?