The simplest and most straightforward type of proof is a
\direct" proof, which we'll call

any proof that follows straight from the definitions or from a direct
calculation . Here's a

couple of examples:

First we'll prove the following statement:

The sum of any two rational numbers is rational.

This proof follows directly from the definition of what it means for a number to
be rational .

**Given that:** r and s are rational numbers.

**Show that:** r + s is rational.

**Proof:** Since r and s are rational, we can write r = p/q and s = m/n for

some integers p, q, m, and n.

Then
.

Since p, q, m, and n are integers, pn+qm and qn are also integers ( the sum

or product of integers is an integer). Thus by the calculation above , r +s is

the quotient of two integers , and is therefore a rational number .

Here's an example of a proof that is really just a calculation. Given the
trigonometric

identity sin(x + y) = sin x cos y + cos x sin y, we'll prove the identity:

sin(2x) = 2 sin x cos x

**Given that:** sin(x + y) = sin x cos y + cos x sin y.

**Show that:** sin(2x) = 2 sin x cos x.

**Proof:**

sin(2x) = sin(x + x)

= sin x cos x + cos x sin x (by the sum
identity )

= 2 sin x cos x:

Note that this proof is merely a string of equalities connecting sin(2x) to 2
sin x cos x. Many

proofs are like this ; when proving an identity or an equation we often start
from one side

of the equation and work towards the other.