6.2. Pythagorean Identities http://www.ck12.org
Cofunctions are not always connected directly through a Pythagorean identity.
tan^2 x+cot^2 x 6 = 1
Visually, the right triangle connecting tangent and secant can also be observed in the unit circle. Most people do
not know that tangent is named “tangent” because it refers to the distance of the line tangent from the point on the
unit circle to thexaxis. Look at the picture below and think about why it makes sense that tanxand secxare as
marked. tanx=o p pad j. Since the adjacent side is equal to 1 (the radius of the circle), tanxsimply equals the opposite
side. Similar logic can explain the placement of secx.
Vocabulary
ThePythagorean Theoremstates that the sum of the squares of the two legs in a right triangle will always be the
square of the hypotenuse.
ThePythagorean Identitystates that since sine and cosine are equal to two legs in a right triangle with a hypotenuse
of 1, then their relationship is that of the Pythagorean Theorem.
Guided Practice
- Derive the following Pythagorean identity:
tan^2 x+ 1 =sec^2 x - Simplify the following expression.
(sec^2 x)( 1 −sin^2 x)−(sincscxx+cossecxx) - Simplify the following expression.
(cost−sint)^2 +(cost+sint)^2
Answers: