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.