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The Pythagorean Identities

You are going to need to quickly recall the three Pythagorean Identities.  The first one is easy to remember because it's just the Pythagorean Theorem.

a^2 + b^2 = c^2

on the unit circle.

 right triangle in the unit circle

a = sin( theta )   b = cos( theta )   c = 1

sin^2( theta ) + cos^2( theta ) = 1

 

But, can you remember the other two?  If you forget, here's the quick way to get them from the first one:


[ sin^2( theta ) / sin^2( theta ) ] + [ cos^2( theta ) / sin^2( theta ) ] = 1 / sin^( theta )

1 + cot^2( theta ) = csc^2( theta )

(You can also remember that the "co" guys go together!)
 


[ sin^2( theta ) / cos^2( theta ) ] + [ cos^2( theta ) / cos^2( theta ) ] = 1 / cos^2( theta )

cot^2( theta ) + 1 = csc^2( theta )


sin^2( theta ) + cos^2( theta ) = 1
Let this one guide you...

1 / sin^2( theta ) = csc^2( theta )
So, if you want the guy with a
csc^2( theta ) , divide by sin^2( theta ) .
1 / cos^2( theta ) = sec^2( theta )
So, if you want the guy with a
sec^2( theta ) , divide by cos^2( theta ) .

You'll still be doing a lot of simplifying of trig expressions in Calculus, and these come up a lot!

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