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The Cofunction and Odd-Even Identities

The Cofunction Identities

sin ( π 2 x ) = cos ( x ) cos ( π 2 x ) = sin ( x ) tan ( π 2 x ) = cot ( x ) csc ( π 2 x ) = sec ( x ) sec ( π 2 x ) = csc ( x ) cot ( π 2 x ) = tan ( x )

Example:

Find the value of cot ( 60 ° ) .

Use the co-function identity tan ( 90 ° θ ) = cot ( θ ) to rewrite the problem.

cot ( 60 ° ) = tan ( 90 ° 60 ° ) = tan ( 30 ° ) = 3 3


The Odd-Even Identities

cos ( x ) is an even function, sin ( x ) is an odd function as trigonometric functions for real variables.

sin ( x ) = sin ( x ) cos ( x ) = cos ( x ) tan ( x ) = tan ( x ) csc ( x ) = csc ( x ) sec ( x ) = sec ( x ) cot ( x ) = cot ( x )

Example:

If cos ( α ) = 1 2 , determine cos ( α ) and sec ( α ) .

cos ( α ) = cos ( α ) = 1 2

sec ( α ) = 1 cos ( α ) = 1 ( 1 2 ) = 2

sec ( α ) = sec ( α ) = 2