CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 6. Analytic Trigonometry

cos^2 x+sin^2 x= 1
Most people rewrite the order of the sine and cosine so that the sine comes first.
sin^2 x+cos^2 x= 1
The two other Pythagorean identities are:

• 1+cot^2 x=csc^2 x


• tan^2 x+ 1 =sec^2 x

To derive these two Pythagorean identities, divide the original Pythagorean identity by sin^2 xand cos^2 xrespectively.
Example A
Derive the following Pythagorean identity: 1+cot^2 x=csc^2 x.
Solution: First start with the original Pythagorean identity and then divide through by sin^2 xand simplify.

sin^2 x

sin^2 x+

cos^2 x

sin^2 x=

1

sin^2 x

1 +cot^2 x=csc^2 x

Example B
Simplify the following expression:sinx( 1 csc−sinx−xsinx).
Solution:

sinx(cscx−sinx)

1 −sinx =

sinx·cscx−sin^2 x

1 −sinx

=^1 −sin

(^2) x
1 −sinx
=(^1 −sin 1 −x)(sin^1 +xsinx)
= 1 +sinx
Note that factoring the Pythagorean identity is one of the most powerful applications. This is very common and is
a technique that you should feel comfortable using.
Example C
Prove the following trigonometric identity:(sec^2 x+csc^2 x)−(tan^2 x+cot^2 x) =2.
Solution: Group the terms and apply a different form of the second two Pythagorean identities which are 1+cot^2 x=
csc^2 xand tan^2 x+ 1 =sec^2 x.
(sec^2 x+csc^2 x)−(tan^2 x+cot^2 x) =sec^2 x−tan^2 x+csc^2 x−cot^2 x
= 1 + 1
= 2
Concept Problem Revisited