University Maths Solution

Maths Question
Question 1

$\text{Show that }\sin 8x+\sin 10x=2\sin 9x\cos x$

Question 2

$\text{Show that }\sin 6x-\sin 2x=2\cos 4x\sin 2x$

Question 3

$\text{Show that }\frac{\sin \theta +\sin \phi }{\cos \theta +\cos \phi }=\tan \left( \frac{\theta +\phi }{2} \right)$

Question 4

$\text{Show that }\frac{\cos 2\alpha +\cos 5\alpha +\cos 8\alpha }{\sin 2\alpha +\text{sin}5\alpha +\sin 8\alpha }=\cot 5\alpha $

Question 5

$\begin{align}  & \text{If }A,B,C\,\text{are angles  of the triangle prove that } \\ & {{(\sin B-\cos B)}^{2}}+{{(\sin C-\cos C)}^{2}}-{{(\sin A-\cos A)}^{2}}=1-4\sin A\cos B\cos C \\\end{align}$

Question 6

$\text{Prove that in a triangle }ABC\text{ }\tan \left( \frac{B-C}{2} \right)=\frac{b-c}{b+c}\cot \frac{A}{2}$