Question

Please help me to prove this..​

Answer 1

Answer: see proof below

Explanation:

Use the following Sum to Product Identities:

`\sin x+\sin y=2\sin\bigg((x+y)/(2)\bigg)\cos \bigg((x-y)/(2)\bigg)\\\\\\\cos x-\cos y=-2\sin\bigg((x+y)/(2)\bigg)\sin \bigg((x-y)/(2)\bigg)`

Proof LHS → RHS

`\text{LHS:}\qquad \qquad \qquad \qquad (\sin \theta-\sin 3\theta+\sin 5\theta -\sin 7\theta)/(\cos \theta -\cos 3\theta -\cos 5\theta +\cos 7\theta)`

`\text{Regroup:}\qquad \qquad \qquad ((\sin 5\theta+\sin \theta)-(\sin 7\theta +\sin 3\theta))/(-(\cos 5\theta -\cos \theta) +(\cos 7\theta -\cos 3\theta))`

`\text{Sum to Product:}\quad (2\sin\bigg((5\theta +\theta)/(2)\bigg)\cos \bigg((5\theta -\theta)/(2)\bigg)-2\sin \bigg((7\theta + 3\theta)/(2)\bigg)\cos \bigg((7\theta - 3\theta)/(2)\bigg))/(2\sin\bigg((5\theta +\theta)/(2)\bigg)\sin \bigg((5\theta -\theta)/(2)\bigg)-2\sin \bigg((7\theta + 3\theta)/(2)\bigg)\sin \bigg((7\theta - 3\theta)/(2)\bigg))`

`\text{Simplify:}\qquad \qquad (2\sin 3\theta \cos 2\theta-2\sin 5\theta \cos 2\theta)/(2\sin 3\theta \sin 2\theta-2\sin 5\theta \cos 2\theta)`

`\text{Factor:}\qquad \qquad (\cos 2\theta(2\sin 3\theta -2\sin 5\theta \cos 2\theta))/(\sin 2\theta(2\sin 3\theta -2\sin 5\theta \cos 2\theta))`

`\text{Simplify:}\qquad \qquad (\cos 2\theta)/(\sin 2\theta)\\\\.\qquad \qquad \qquad =\cot 2\theta`

LHS = RHS: cot 2Ф = cot 2Ф
`\checkmark`