## Trig Identities

### Algebra: Trigonometry Help

#### Home | Trig menu | Algebra menu

If we start off with a right angle triangle with a hypotenuse of length 1, we can use the Pythagorean theorem to get

x^{2} + y^{2} =
1

But, since

cos a = x/1 = x

sin a = y/1 = y

we can substitute to get

cos^{2}a +
sin^{2}a = 1

The other things to remember are that

On top of that, it helps to know

cos(2a) =
cos^{2}a -
sin^{2}a

sin(2a) =
2(cos a)
(sin a)

stemming from the facts that

cos(a + b) =
cos(a)cos(b) - sin(a)sin(b)

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Knowing these basic facts will help you solve any trig identity
problems you might encounter. For example, if we take the basic squared
equation above and divide both sides by
cos^{2}a we get

1 + tan^{2}a =
sec^{2}a

Similarly, if we divide the basic squared equation by
sin^{2}a we get

cot^{2}a + 1
= csc^{2}a

In solving trig identity problems, I usually find that converting everything to sine and cosine keeps my life simpler. It also helps to know how to multiply and divide fractions.

**Example 1:** Prove sin^{2}a +
cos^{2}a =
(sina)(csca)

On the left hand side, we know that it equals 1. All we have to do is show that the right hand side is 1 as well. We know that csc = 1/sin. So,

(sina)(csca)

= (sina) x 1/(sina)

= (sina)/(sina)
= 1

Sometimes, a little bit of intuition comes in handy, especially when you're missing something on one side of the equation that you have on the other. This next example is a case in point.

**Example 2:** Prove 2cos^{2}a
= cos(2a) + 1

There's not much we can do with the left hand side. So, let's expand the right hand side.

cos(2a) + 1

= cos^{2}a -
sin^{2}a + 1

Isn't that wonderful? We've got a sin^{2} on the right hand
side but not on the left. And what do we do with that 1 on the right
hand side? This is where knowing your basic trig identities comes to the
rescue. We can use the fact that **sin ^{2} + cos^{2} =
1.**

cos^{2}a -
sin^{2}a + 1

= cos^{2}a -
sin^{2}a +
sin^{2}a +
cos^{2}a

= cos^{2}a +
cos^{2}a

= 2cos^{2}a

which is the same as the left hand side. Notice that in this problem, we didn't touch the left hand side at all.

This last example illustrates the point that converting everything to sin and cos and knowing how to multiply and divide fractions can make your life easier.

**Example 3:** Prove (csc a)/
(cot^{2}a) =
(tan a)(sec a).

They don't get much uglier than this do they? Well, lets start with the left hand side.

(csc a)/
(cot^{2}a)

= (1/sin a)/
[(cos^{2}a)/
(sin^{2}a)]

= = (1/sin a) x
[(sin^{2}a)/
(cos^{2}a)]

= (sin^{2}a)/
[(sin a)
(cos^{2}a)]

= (sin a)/
(cos^{2}a)

which is as far as we can go with the left hand side. Now the right hand side.

(tan a)(sec a)

= (sin a)/(cos a)
x 1/(cos a)

= (sin a)/
(cos^{2}a)

and the two sides are the same.

## Algebra: Trigonometry Help

### What do you need help with? Click on one of the following topics...

Copyright © 2011 - Mathwizz.com all rights reserved