Question

Show that none of the following is an identity:

`tan^2 theta + sin theta = cos^2 theta`

Answer 1

`tan^2 theta + sin theta = cos^2 theta`

 LHS = `tan^2 theta + sin theta `

       =`(sin^2 theta)/(cos^2 theta) + sin theta`

      =` (1- cos^2 theta )/( cos^2 theta) + sin theta`

      =` sec^2 theta  -1 + sin theta `

  Since LHS ≠ RHS, this is not an identity.