Proofs

From the Pythagorean identity, sin2x = 1-cos2x.

LHS = 1/(sinx cosx) - cosx/sinx

LHS = 1/(sinx cosx) - (cosx/sinx)(cosx/cosx)

LHS = 1/(sinx cosx) - cos2x/(sinx cosx)

LHS = (1- cos2x)/(sinx cosx)

LHS = sin2x /(sinx cosx) [from Pythagorean identity]

LHS = sin2x /(sinx cosx)

LHS = sinx/cosx

LHS = tanx [by definition]

RHS = tanx

LHS = RHS and so the identity is proven.

Q.E.D.

positivi

Yes, they do.

rhombus

You can convert this to base ten by re-writing 3096 as a summation of hex powers:

3*16^3 + 0*16^2 + 9*16^1 + 5*16^0 = 12437 in base 10

False. G may be a finite group without sub-groups.

A zillion is ultra god money

Six faces.

DeMorgan's theorem can be simply be stated simply as "Break the line, and change the sign." Where line represents complement...

10:2

Assume that f:S->T is invertible with inverse g:T->S, then by definition of invertible mappings f*g=i(S) and g*f=i(T), which defines f as the inverse of g. So g is invertible.

every abelian group is not cyclic. e.g, set of (Q,+) it is an abelian group but not cyclic.

Yes, 1 and 107.

58

The buoyant force is equal to the weight of the displaced fluid (Archimedes Principle). A gallon jug underwater will have a buoyant force equal to the weight of the displaced water: eight pounds.

The total weight will also include the normal downward weight of the jug itself and the air it contains.

Use the law of sines.

The trivial subsets of a set are those subsets which can be found without knowing the contents of the set.

The empty set has one trivial subset: the empty set.

Every nonempty set S has two distinct trivial subsets: S and the empty set.

Explanation:

This is due to the following two facts which follow from the definition of subset:

Fact 1: Every set is a subset of itself.

Fact 2: The empty set is subset of every set.

The definition of subset says that if every element of A is also a member of B then A is a subset of B. If A is the empty set then every element of A (all 0 of them) are members of B trivially. If A = B then A is a subset of B because each element of A is a member of A trivially.

As I understand the question: yes, f(x) can be a function even if f(x) is not defined for all x. For example, f(x) = x/x is a function that is equal to 1 everywhere but at x=0, where it is undefined.

The kind of rock you see forming in Puu Oo crater is extrusive igneous. This type of rock is formed by lava.

It is 959334055936.

Divide by 100. ie move decimal point 2 spots to the left, so 158cm becomes 1.58m

False

Show that sec'x = d/dx (sec x) = sec x tan x.

First, take note that

sec x = 1/cos x;

d sin x = cos x dx;

d cos x = -sin x dx; and

d log u = du/u.

From the last, we have du = u d log u.

Then, letting u = sec x, we have,

d sec x = sec x d log sec x; and

d log sec x = d log ( 1 / cos x )

= -d log cos x

= d ( -cos x ) / cos x

= sin x dx / cos x

= tan x dx.

Thence, d sec x = sec x tan x dx, and

sec' x = sec x tan x,

which is what we set out to show.