## Trigonometry 7th Edition

With $\displaystyle \csc\theta=\frac{1}{\sin\theta},\quad$and $\sin(-\theta)=-\sin\theta$ $LHS=\displaystyle \frac{1}{\sin\theta}-\sin\theta,\quad$ ... common denominator is $\sin \theta$ $LHS=\displaystyle \frac{1-\sin^{2}\theta}{\sin\theta}$ in the numerator, apply the Pythagorean identity $\cos^{2}\theta+\sin^{2}\theta=1$ $LHS=\displaystyle \frac{\cos^{2}\theta+\sin^{2}\theta-\sin^{2}\theta}{\sin\theta}=\frac{\cos^{2}\theta}{\sin\theta}=RHS$