Question

Convert the following sum into a product. cosx + cos3x + cos5x + cos7x

Answer 1

`4\cos x\cos (4x)\cos (2x)`

Explanation:

Given:
`cosx + cos3x + cos5x + cos7x`

To convert: the given sum into product

Solution:

Use formula:
`\cos A+\cos B=2\cos \left ( (A+B)/(2) \right )\cos \left ( (A-B)/(2) \right )`

`cosx + cos3x + cos5x + cos7x=2\cos \left ( (x+3x)/(2) \right )\cos \left ( (x-3x)/(2) \right )+2\cos \left ( (5x+7x)/(2) \right )\cos \left ( (5x-7x)/(2) \right )\\=2\cos (2x)\cos (-x)+2\cos (6x)\cos (-x)\\=2\cos (2x)\cos (x)+2\cos (6x)\cos (x)\\=2\cos x\left [ \cos (2x)+\cos (6x) \right ]`

`cosx + cos3x + cos5x + cos7x=2\cos \left ( (x+3x)/(2) \right )\cos \left ( (x-3x)/(2) \right )+2\cos \left ( (5x+7x)/(2) \right )\cos \left ( (5x-7x)/(2) \right )\\=2\cos x\left [ \cos (2x)+\cos (6x) \right ]\\=2\cos x\left [2 \cos \left ( (2x+6x)/(2) \right )\cos \left ( (2x-6x)/(2) \right ) \right ]\\=2\cos x\left [ 2\cos (4x) \cos (-2x) \right ]\\=4\cos x\cos (4x)\cos (2x)`