What textbooks should I use for Trigonometry and Calculus? My basics are terrible. I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by memorizing formulas just before an exam. I like mathematics but I am terrible at learning these identities.
I have one paper coming up and need a lot of practice. These are the topics that is going to be covered in the paper:
Differential calculus:

Definition of the limit of a function in ε-δ form- Algebra of
  limits-Continuity of a function - types of discontinuities -properties
  of continuous functions on a closed interval (boundedness, attainment
  of bounds and taking every value between bounds) - differentiability,
  differentiability implies continuity and converse is not true. Rolle’s
  theorem-Lagrange’s and Cauchy’s first mean value theorems - Taylors’
  theorem with Lagrange’s form of remainder - Maclaurin’s expansion-
  problems. Evaluation of limits by L’Hospital’s rule (indeterminate
  form).

Group Theory:

Recapitulation of the definition and standard properties of groups and
  subgroups Cyclic groups- properties-order of an element of a group-
  properties related to order of an elementsubgroup generated by an
  element of a group- coset decomposition of a group- modulus relation-
  index of a group- Lagrange's theorem for finite groups-consequences. 

Sequences:

Definition of a sequence-limit of a sequence-algebre of limits.
  Convergent, divergent and oscillatory sequence-infimum-supremum-Nature
  of the sequences theorems-problems-monotonicity-problems-cauchy
  sequences.

Series:

Definition of Convergence-divergence-oscillation-properties of
  convergent series- Cauchy’s theorem-geometric series - p-series
  -comparison tests, De'alembert's test, Raabe's tests. Cauchy root test - problems. (The following tests are without proof) Absolute-conditional convergence-De'alembert's test for absolute
  convergence Alternating series. Summation of series - binomial,
  exponential and logarithmic series.

Differential Equations:

i. Formation and solution of ordinary differential equations (I order
  & I degree). a) Variable-separable and reducible to variable separable
  form b) Homogeneous & reducible to homogeneous forms c) Linear
  equations- Bernoulli’s equations and those reducible to these
  equations d) Exact equations and reducible to exact with standard
  integrating factors ii. Equation of first order and higher
  degree-Clairaut’s equation -general and singular solutions- geometric
  meaning. iii. Orthogonal trajectories in cartesian and polar forms.

How should I go about learning the basics for these topics in order to understand the syllabus better?
Thank you so much.
 A: Many trig formulas are things you just look up (or memorize).  They can also be sometimes be derived from each other.  For example, knowing $\sin^2 x + \cos^2 x =1$, dividing by $\cos^2 x$ yields $\tan^2 x + 1 = \sec^2 x.$  A useful exercise would be to take a list of rules (maybe from your textbook) and see which ones can be derived from other ones.  It seems like you shouldn't spend too much time on trig, as long as you know what the 6 trig functions mean and how they're related to each other.
Now for some general comments on preparing for your exam:  Of course you want to work on problems given to you, working your way from getting assistance towards doing the problem without referring to a textbook/the internet.
Something that may help with forgetting things when the exam comes, is to choose a topic and come up with examples to play with. For example in group theory, take your specimens to be the integers mod 3, the integers mod 6, the integers mod 9, and the integers mod 12 (I'm just picking a variety of things).  Now you can verify that these are groups by checking the properties.  What are all of their subgroups?  What are the cosets of those subgroups?  What is everything that Lagrange's theorem tells you about each of the groups?  Compare and contrast the answers of all of these questions for the different groups.
I assume you have a textbook or some sort of resource from the time you learned these things to begin with.  Most textbooks are probably fine as long as they cover the material you listed and have lots of practice problems.
A: for basics of trigonometry you can go for S.L. lony bok,
and the topics what you have mentioned i will suggest two books 
1) advanced engineering mathematics by erwin kreyszig
2) advanced engineering mathematics by ramana
the first one is for deep understanding of the topics and second one is for practice. 
also if you are still looking for further basics in this topics you can try any demystify maths 
all the best and enjoy reading 
regards 
chirag
