Question

Please someone help me... i need full answer ​

Answer 1

Answer: see proof below

Explanation:

Use the following Sum to Product Identities:

`\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)`

Proof LHS → RHS

`\text{LHS:}\qquad \qquad \qquad \qquad (\sin 2A+\sin 5A-\sin A)/(\cos A+\cos 2A+\cos 5A)`

`\text{Regroup:}\qquad \qquad \qquad (\sin 2A+(\sin 5A-\sin A))/(\cos 2A+(\cos 5A+\cos A))`

`\text{Sum to Product:}\qquad (\sin 2A+2\cos\bigg((5A+A)/(2)\bigg)\sin\bigg((5A-A)/(2)\bigg))/(\cos2A+\cos\bigg((5A+A)/(2)\bigg)\cos\bigg((5A-A)/(2)\bigg))`

`\text{Simplify:}\qquad \qquad \qquad (\sin 2A+2\cos 3A\sin 2A)/(\cos 2A+2\cos 3A\cos 2A)`

`\text{Factor:}\qquad \qquad \qquad (\sin 2A(1+2\cos 3A))/(\cos 2A(1+2\cos 3A))`

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

LHS = RHS: tan 2A = tan 2A
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