Question

Please someone 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)\\\\\\\sin x-\sin y=2\cos \bigg((x+y)/(2)\bigg)\sin \bigg((x-y)/(2)\bigg)\\\\\\\cos x+\cos y=2\cos \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 (\sin 5-\sin 15+\sin 25 - \sin 35)/(\cos 5-\cos 15- \cos 25 + \cos 35)`

`\text{Reqroup:}\qquad \qquad \qquad ((\sin 25+\sin 5)-(\sin 35 + \sin 15))/((\cos 35+\cos 5)-(\cos 25 + \cos 15))`

`\text{Sum to Product:}\quad (2\sin \bigg((25+5)/(2)\bigg)\cos \bigg((25-5)/(2)\bigg)-2\sin \bigg((35+15)/(2)\bigg)\cos \bigg((35-15)/(2)\bigg))/(2\cos \bigg((25+15)/(2)\bigg)\cos \bigg((25-15)/(2)\bigg)-2\cos \bigg((35+5)/(2)\bigg)\cos \bigg((35-5)/(2)\bigg))`
`\text{Simplify:}\qquad \qquad (2\sin 15\cos 10-2\sin 25\cos 10)/(2\cos 20\cos 15-2\cos 20\cos 5)`

`\text{Factor:}\qquad \qquad (2\cos 10(\sin 15-\sin 25))/(2\cos 20(\cos 15-\cos 5))`

`\text{Sum to Product:}\qquad (\cos 10\bigg[2\cos \bigg((15+25)/(2)\bigg)\sin \bigg((15-25)/(2)\bigg)\bigg])/(\cos 20\bigg[-2\sin \bigg((15+5)/(2)\bigg)\sin \bigg((15-5)/(2)\bigg)\bigg])`

`\text{Simplify:}\qquad \qquad (\cos 10[2\cos 20\sin (-5)])/(\cos 20[-2\sin 10\sin 5])\\\\\\.\qquad \qquad \qquad =(-2\cos10 \cos 20 \sin 5)/(-2\sin 10 \cos 20 \sin 5)\\\\\\.\qquad \qquad \qquad =(\cos 10)/(\sin 10)\\\\\\.\qquad \qquad \qquad =\cot 10`

LHS = RHS: cot 10 = cot 10
`\checkmark`