Question

Can you express the following as a product using the sum-to-product formula?

cos(x)+2cos(7x)+cos(13x)

Answer 1

`\large\underline{\mathbb{SOLUTION:}}`

We use the Sum-To-Product formula on this case:

`\small\boxed{\tt{ \cos (A) + \cos (B)=2 \cos ( (A + B)/(2) ) \: \cdot \: (\cos(A - B)/(2))}}`

Where,

`\bullet\longrightarrow \tt{A = \purple{x }}`

`\bullet \longrightarrow{ \tt{B = \purple{13x}}}`

Then substitute those values to calculate and express the following as a product:

`\tt{\longrightarrow2 \cos (7x) + 2 \cos ( (x + 13)/(2) ) \: \cdot \: \cos((x - 13x)/(2))}`

`\purple \\ \tt{\longrightarrow 2 \cos (7x) + 2 \cos ( \frac{ \bold{14x}}{2} ) \: \cdot \: \cos( ( - 12x)/(2) )}`

`\boxed{\small{\tt{\purple{ \longrightarrow = 2 \cos(7x) + 2 \cos(7x) \cdot( - 6)}}}}`

`\large\underline{\mathbb{ANSWER:}}`

Therefore, the final answer is
`\small{\rm{ 2 \cos(7x) + 2 \cos(7x) \cdot( - 6)}}`