Trig without Tears Part 6:

# The “Squared” Identities

Copyright © 1997–2020 Stan Brown, BrownMath.com

**Summary:**
This chapter begins exploring trigonometric identities.
Three of them involve only **squares of functions**.
These are called **Pythagorean identities** because they’re just
the good old Theorem of Pythagoras in new clothes. Learn
**the really basic one**, namely
sin² *A* + cos² *A* = 1, and
the others **are easy to derive** from it in a single step.

Students seem to get bogged down in the huge number of trigonometric identities. As I said earlier, I think the problem is that students are expected to memorize all of them. But really you don’t have to, because they’re all just forms of a very few basic identities. The next couple of chapters will explore that idea.

For example, let’s start with the really basic identity:

(38) sin² *A* + cos² *A* = 1

That one’s easy to remember: it involves only the basic sine and cosine, and you can’t get the order wrong unless you try.

But you don’t have to remember even that one, since it’s really just another
form of the **Pythagorean Theorem**. (You do remember *that,* I
hope?) Just think about a right triangle with a hypotenuse of one
unit, as shown at right.

First convince yourself that the figure is right, that the lengths
of the two legs are sin *A* and cos *A*. (Check back in the
section on lengths of sides, if you
need to.)
Now write down the
Pythagorean Theorem for this triangle. Voilà!
You’ve got equation 38.

What’s nice is that you can get the other “squared” or
**Pythagorean identities** from this one, and you don’t have to
memorize any of them. Just start with equation 38 and divide
through by either sin² *A* or cos² *A*.

For example, what about the riddle we started
with, the relation between
tan² *A* and sec² *A*? It’s easy to answer by a quick
derivation—easier than memorizing, in my opinion.

If you want an identity involving tan² *A*, remember
equation 3: tan *A* is defined to be sin *A*/cos *A*.
Therefore, to create an identity involving tan² *A* you need
sin² *A*/cos² *A*. So take equation 38 and divide through
by cos² *A*:

sin² *A* + cos² *A* = 1

sin² *A*/cos² *A* + cos² *A*/cos² *A* =
1/cos² *A*

(sin *A*/cos *A*)² + 1 = (1/cos *A*)²

which leads immediately to the final form:

(39) tan² *A* + 1 = sec² *A*

You should be able to work out the third identity (involving
cot² *A* and csc² *A*) easily enough. You
can either start with equation 39 above and use
the cofunction rules (equation 6 and
equation 7), or
start with equation 38
and divide by something appropriate. Either way, check to make sure
that you get

(40) cot² *A* + 1 = csc² *A*

It may be easier for you to visualize these
two identities geometrically. Start with the sin *A*,
cos *A*, 1 right triangle above. Divide all three sides by cos *A*
and you get the first triangle below; divide by sin *A* instead
and you get the second one. You can then just read off the Pythagorean
identities.

From the first triangle,
tan² *A* + 1 = sec² *A*; from the second
triangle,
cot² *A* + 1 = csc² *A*.

To get the most benefit from these problems, work them without first looking at the solutions. Refer back to the chapter text if you need to refresh your memory.

**Recommendation**: Work them on paper —
it’s harder to fool yourself about whether you really
understand a problem completely.

You’ll find full solutions for all problems. Don’t just check your answers, but check your method too.

1
If sin *A* = 3/4, find cos *A*.

2
tan *B* = −2√2. Find sec *B*.

3
tan *C* = √15. Find cos *C*.

4
tan *D* =√15, Find sin *D*.

5Prove:
sin² *x* = tan² *x* / (tan² *x* + 1)

This assumes that*x* ≠
π/2 + *k*π, for integer
*k*—or 90° + 180*k*°, if you
prefer—because the tangent is undefined for those angles.

This assumes that

**20 Nov 2016**: Added practice problems. Simplified the last couple of paragraphs, about visualizing equations 39 and 40 geometrically.**1 Nov 2016**: Updated the mathematical notation, particularly the use of italics and spaces, to conform to the standard. I used Jukka Korpela’s comprehensive Writing Mathematical Expressions (2014, Suomen E-painos Oy), ISBN 978-952-6613-25-3.- (intervening changes suppressed)
**19 Feb 1997**: New document.

next: 7/Sum and Difference

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