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nadya68 [22]
2 years ago
10

Help please!

Mathematics
1 answer:
KengaRu [80]2 years ago
3 0

Answer:  see proof below

<u>Step-by-step explanation:</u>

Use the following Sum to Product Identities:

\cos A+\cos B=2\cos \bigg(\dfrac{A+B}{2}\bigg)\cdot \cos \bigg(\dfrac{A-B}{2}\bigg)\\\\\\\sin A-\sin B=2\cos \bigg(\dfrac{A+B}{2}\bigg)\cdot \sin \bigg(\dfrac{A-B}{2}\bigg)

Use Even/Odd Identities:    cos (-A) = - cos A

                                             sin (-A) =   sin (A)

Use Quotient Identity:   \cot A=\dfrac{\cos A}{\sin A}

<u>Proof LHS →  RHS:</u>

\text{LHS:}\qquad \qquad \qquad \dfrac{\cos x+\cos 7x}{\sin x - \sin 7x}

\text{Sum to Product:}\qquad \dfrac{2\cos \bigg(\dfrac{x+7x}{2}\bigg)\cdot \cos \bigg(\dfrac{x-7x}{2}\bigg)}{2\cos \bigg(\dfrac{x+7x}{2}\bigg)\cdot \sin \bigg(\dfrac{x-7x}{2}\bigg)}

\text{Simplify:}\qquad \qquad \quad \dfrac{2\cos \bigg(\dfrac{8x}{2}\bigg)\cdot \cos \bigg(\dfrac{-6x}{2}\bigg)}{2\cos \bigg(\dfrac{8x}{2}\bigg)\cdot \sin \bigg(\dfrac{-6x}{2}\bigg)}

                          = \dfrac{ \cos (-3x)}{\sin(-3x)}

\text{Even Odd:}\qquad \quad  \dfrac{ -\cos (3x)}{\sin(3x)}

\text{Quotient:}\qquad \quad -\cot(3x)

LHS = RHS  \checkmark

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