91Ó°ÊÓ

In any triangle, if \(\tan \frac{A}{2}=\frac{5}{6}\) and \(\tan \frac{B}{2}=\frac{20}{37}\), find \(\tan \frac{C}{2}\) and prove that in this triangle \(a+c=2 b\).

In a \(\Delta A B C, a=13, b=14, c=15\), then find \(\sin \frac{A}{2}\).

If \(a, b\) and \(c\) be in A.P. prove that i. \(\cot \frac{A}{2}, \cot \frac{B}{2}\) and \(\cot \frac{C}{2}\) are in A.P. ii. \(\cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}\) and \(\cos C \cot \frac{C}{2}\) are in A.P. iii. \(a \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{A}{2}=\frac{3 b}{2}\). iv. \(\tan \frac{A}{2}+\tan \frac{C}{2}=\frac{2}{3} \cot \frac{B}{2}\). v. \(\cot \frac{A}{2} \cot \frac{C}{2}=3\).

If \(a, b\) and \(c\) are in H.P., prove that \(\sin ^{2} \frac{A}{2}, \sin ^{2} \frac{B}{2}\) and \(\sin ^{2} \frac{C}{2}\) are also in H.P.

The sides of a triangle are in A.P. and the greatest and least angles are \(\theta\) and \(\phi\); prove that \(4(1-\cos \theta)(1-\cos \phi)=\cos \theta+\cos \phi\)

The sides of a triangle are in A.P. and the greatest angle exceeds the least by \(90^{\circ}\); prove that the sides are proportional to \(\sqrt{7}+1, \sqrt{7}\) and \(\sqrt{7}-1\).

If \(C=60^{\circ}\), then prove that \(\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}\).

In any triangle prove that, if \(\theta\) be any angle, then \(b \cos \theta=c \cos (A-\theta)+a \cos (C+\theta)\).

If \((a+b+c)(b+c-a)=k b c\), then prove that \(k \in(0,4)\).

The sides of a triangle are \(a, b, \sqrt{a^{2}+a b+b^{2}}\), prove that the greatest angle is \(120^{\circ} .\)