Solution of Triangle

Form the pair of linear equations for the following problems and find their solution by the substitution method.
 The larger of two supplementary angles exceeds the smaller by $$18$$ degrees. Find them
In a $$\triangle ABC,$$ right angled at $$A$$. The radius of the inscribed circle is $$2$$cm.Radius of the circle touching the side $$BC$$ and also sides $$AB$$ and $$AC$$ produced is $$15$$cm.The length of the side $$BC$$ measured in cm is
The altitude of a triangle is two-third the length of its corresponding base. If the altitude is increased by 4 cm and the base is decreased by 2 cm ,the area of the triangle remains the same. Find the base and the altitude of the triangle.
In any $$\triangle ABC, \dfrac{\cos{2A}}{{a}^{2}}-\dfrac{\cos{2B}}{{b}^{2}}$$ is independent of the measures of the angles $$A$$ and $$B$$
For any $$\triangle ABC$$,
$$\cos^{2}\left(\dfrac {A}{2}\right) + \cos^{2}\left(\dfrac {B}{2}\right)+ \cos^{2}\left(\dfrac {C}{2}\right)=2\sin\left(\dfrac {A}{2}\right) 2+2\sin\left(\dfrac {B}{2}\right) \sin\left(\dfrac {C}{2}\right)$$