Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 37218 by Rio Mike last updated on 10/Jun/18

$$Proovethat$$$$a)\frac{1-cosA+cosB-cos(A+B)}{1+cosA-cosB-cos(A+B)}=tan\frac{A}{2}.cot\frac{B}{2}$$$$b)cos\alpha cos(60-\alpha )cos(60+\alpha )=\frac{1}{4}cos3\alpha$$$$$$

Answered by $@ty@m last updated on 11/Jun/18

$$\left(a\right)L.H.S.$$$$=\frac{1-cosA+cosB-cos(A+B)}{1+cosA-cosB-cos(A+B)}$$$$=\frac{1-2\sin \frac{A+B}{2}\sin \frac{B-A}{2}-1+{2\sin }^2\frac{A+B}{2}}{1+2\sin \frac{A+B}{2}\sin \frac{B-A}{2}-1+{2\sin }^2\frac{A+B}{2}}$$$$=\frac{\sin \frac{A-B}{2}+\sin \frac{A+B}{2}}{\sin \frac{B-A}{2}+\sin \frac{A+B}{2}}$$$$=\frac{2\sin \frac{A}{2}\cos \frac{B}{2}}{2\sin \frac{B}{2}\cos \frac{A}{2}}$$$$=\tan \frac{A}{2}\cot \frac{B}{2}$$$$=R.H.S.$$$$\left(b\right)L.H.S.$$$$=cos\alpha cos(60-\alpha )cos(60+\alpha )$$$$=\frac{1}{2}.\cos \alpha .2cos(60-\alpha )cos(60+\alpha )$$$$=\frac{1}{2}.\cos \alpha .(\cos 120+\cos 2\alpha )$$$$=\frac{1}{2}.\cos \alpha .(-\frac{1}{2}+2\cos {}^2\alpha -1)$$$$=\frac{1}{2}.\cos \alpha .(2\cos {}^2\alpha -\frac{3}{2})$$$$=\frac{1}{4}.(4\cos {}^3\alpha -3\cos \alpha )$$$$=\frac{1}{4}\cos 3\alpha$$$$=R.H.S.$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com