Prove the following trigonometric identities:

Question

a) 1+cos2x=2cos²x

 

b) csc²[(PIE/2)-x]-1=tan²x

 

 

Answer 1

a) we know that     sin²x + cos²x = 1     and   cos 2x = cos²x - sin²x

 

therefore,

       1+cos2x = (sin²x + cos²x) +(cos²x - sin²x)

                     = 2cos²x

 

 

b) as csc[(PIE/2)-x] = sec x    and sec²x = 1+ tan²x

 

Therefore,

                  csc²[(PIE/2)-x]-1 =  sec²x - 1

                                            = 1 + tan²x - 1

                                            = tan²x