Last term in the expansion of $\sin 8\theta ,\sin 9\theta$

$\sin 8\theta ={}^8{c}_1.{\cos }^7\theta .\sin \theta -{}^8{c}_3{\cos }^5\theta .{\sin }^3\theta +{}^8{c}_5{\cos }^3\theta {\sin }^5\theta -{}^8{c}_7\cos \theta {\sin }^7\theta$

$=8{\cos }^7\theta \sin \theta -56{\cos }^5\theta {\sin }^3\theta +56{\cos }^3\theta {\sin }^5\theta -8\cos \theta .{\sin }^7\theta$

 $\therefore$The last terms in the expression of $\sin 8\theta =-8\cos \theta {\sin }^7\theta$

 $\begin{array}{l}ii)\sin 9\theta ={}^9{c}_1{\cos }^8\theta \sin \theta -{}^9{c}_3{\cos }^6\theta .{\sin }^3\theta +{}^9{c}_5{\cos }^4\theta .{\sin }^5\theta \\ -{}^9{c}_7{\cos }^2\theta {\sin }^7\theta +{}^9{c}_9{\sin }^9\theta \end{array}$

 $\therefore$The last terms in the expansion of $\sin 9\theta ={\sin }^9\theta$