## Trigonometry (11th Edition) Clone

$$\csc t=\sqrt{1+\cot^2 t}$$ After taking $t=-\frac{\pi}{2}$, we find that $\csc t\ne\sqrt{1+\cot^2t}$ at $t=-\frac{\pi}{2}$. That means the equation is not an identity.
$$\csc t=\sqrt{1+\cot^2 t}$$ We would find an example that disproves the equation, thus showing that it is not an identity. Pick $t=-\frac{\pi}{2}$. That means, $$\csc t=\csc\Big(-\frac{\pi}{2}\Big)=-1$$ and $$\sqrt{1+\cot^2t}=\sqrt{1+\cot^2\Big(-\frac{\pi}{2}\Big)}=\sqrt{1+0^2}=\sqrt1=1$$ $-1\ne1$, which means $\csc t\ne\sqrt{1+\cot^2t}$ at $t=-\frac{\pi}{2}$ The equation is not an identity.