Determine if the following is an identity: $$1+ \sin t = \cos^3 t \tan t + \sin^3 t$$

There is not much to simplify on the left-hand side, so let us start by trying to simplify the right-hand side,$\require{cancel}$

$$\begin{array}{rclc} \cos^3 t \tan t + \sin^3 t &=& \cos^3 t \cdot \frac{\sin t}{\cos t} + \sin^3 t & \scriptsize{\textrm{rewriting in terms of } \sin \theta \textrm{ and } \cos \theta}\\\\ &=& \cos^2 t \; \sin t + \sin^3 t & \scriptsize{\textrm{after canceling } \cos t}\\\\ &=& \sin t \, (\cos^2 t + \sin^2 t) & \overset{\normalsize{\textrm{factoring reveals a hidden value of one}}}{\scriptsize{\textrm{(recall } \cos^2 + \sin^2 = 1 \textrm{)}}}\\\\ &=& \sin t \end{array}$$

Since this is always one less than the left-hand side of the given equation, we know the above equation is not an identity.