Sum, Difference, and Cofunction Identities




                 Sum and Difference Identities for the Cosine
          

          



                              Cofunction Identities
The cosine, cotangent, and cosecant functions are known as cofunctions
of the sine, tangent, and secant functions.

Using the first identities above,

     

                                  
So we have the
                              cofunction identity for the cosine
                                   

Letting      in this equation, we get

          


                         cofunction identity for the sine
                               .

Since

          

we get the
                         cofunction identity for the tangent
                                   


These identities are true for any real number or any angle in radian measure.
If     is in degree measure,
                                                       replace with



                    Sum and Difference Identities for Sine and Tangent

Using the cofunction identity for the cosine, we can write

     


                                .

Simplifying, we get the
                                       Difference Identity for the Sine Function
                              

If we replace    by   , we get the
                                   Sum Identity for the Sine Function
                           

Since the tangent is the ratio of the sine to the cosine, simple algebra gives the

                                  Difference Identity for the Tangent Function
                                   

If we replace by , we get the
                                Sum Identity for the Tangent Function
                                   


See Examples 1 – 5, pages 537 – 540, of the textbook.



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