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### Verifying Trigonometric Identities

```Date: 07/12/2002 at 08:10:05
From: Dianne Graham
Subject: How do you verify a Trigonometry Identity?

How do you verify a Trigonometry Identity?  I looked in my textbook
and the directions were not clear.
```

```
Date: 07/12/2002 at 17:24:22
From: Doctor Ian
Subject: Re: How do you verify a Trigonometry Identity?

Hi Diane,

The idea is to manipulate one or both sides of the identity until
you have something that is obviously true, e.g., sin(x) = sin(x).

It might be instructive to see how you can _create_ an identity.

sin(x) = sin(x)

Now, csc(x) = 1/sin(x), so we can write

sin(x) = 1/csc(x)

And tan(x) = sin(x)/cos(x), so we can write

tan(x)cos(x) = 1/csc(x)

And cos^2(x) + sin^2(x) = 1, so we can write

sin^2(x) + cos^2(x)
tan(x)cos(x) = -------------------
csc(x)

sin^2(x)   cos^2(x)
= -------- + --------
csc(x)     csc(x)

cos^2(x)
= sin^3(x) + --------
csc(x)

Now, this isn't a very elegant identity, but it illustrates
the point:  By starting with a trivially true statement, and
exchanging simple expressions for more complicated equivalent
expressions, you can build up a statement that must be true, but
isn't obviously true.

It's very much like what you do to set up an equation to be
solved in algebra:

http://mathforum.org/library/drmath/view/57265.html

Verifying an identity is a matter of going in the opposite
direction.  Usually you start by converting everything to sines
and cosines:

cos^2(x)
tan(x)cos(x) = sin^3(x) + --------
csc(x)

sin(x)                    cos^2(x)
------cos(x) = sin^3(x) + --------
cos(x)                    1/sin(x)

And then you use the tricks - e.g., factoring, cancellation -
that you were supposed to have learned in algebra:

sin(x) = sin^3(x) + sin(x)cos^2(x)

= sin(x)(sin^2(x) + cos^2(x))

sin(x) = sin(x)

Often the key to verifying an identity is to recognize that
something like

sin(x)cos(y) + cos(x)sin(y),

can be replaced by

sin(x+y)

There's no magic formula for doing these kinds of problems.  It
normally involves a lot of trial and error, since you're
essentially untying a knot that someone else tied, often just
because he thought of a tricky way to tie one.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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