Trigonometry (11th Edition) Clone

We cannot use the method in Example 2 to find a formula for $\tan(270^\circ-\theta)$ because if we do so, we would be stopped halfway when we need a defined value of $\tan270^\circ$ to carry on. However, $\tan270^\circ$ is unfortunately undefined.
To understand why we cannot use the identity of the tangent difference here to find a formula for $\tan(270^\circ-\theta)$, we would use the same identity to expand $\tan(270^\circ-\theta)$ to find out where the problem is. Identity of tangent difference: $$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$ Apply it to $\tan(270^\circ-\theta)$ $$\tan(270^\circ-\theta)=\frac{\tan270^\circ-\tan\theta}{1+\tan270^\circ\tan\theta}$$ The problem lies in here. We cannot go any further, since $\tan270^\circ$ is undefined. The reason why $\tan270^\circ$ could be explained by recalling that $$\tan270^\circ=\frac{\sin270^\circ}{\cos270^\circ}$$ $\sin270^\circ=\sin(-90^\circ)=-\sin90^\circ=-1$ $\cos270^\circ=\cos(-90^\circ)=\cos90^\circ=0$ That means $$\tan270^\circ=\frac{-1}{0}$$ which is obviously undefined. So to conclude, the main problem here is that $\tan270^\circ$ is undefined, meaning the formula cannot be found by using the identity of tangent difference as the value of $\tan270^\circ$ being defined is essential.