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Prove that ("cos " 6x + " cos " 4x...

Prove that `("cos " 6x + " cos " 4x)/("sin " 6x - "sin " 4x) = " cot x`

Text Solution

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We have
`LHS = ("cos " 6x + " cos " 4x)/( " sin " 6x - " sin " 4x)`
`=(2 " cos " ((6x + 4x)/(2)) " cos " ((6x -4x)/(2)))/(2 " cos " ((6x +4x)/(2)) " sin " ((6x -4x)/(2)))`
`{underset("sin " C - "sin " D = 2 " cos " ((C+D)/(2)) " sin " ((C-D)/(2)))( ,:' " cos " C + " cos "D = 2 " cos " ((C+D)/(2)) " cos " ((C-D)/(2))}}`
`=( 2" cos " 5x " cos " x)/(2 " cot " 5x " sin " x) =(" cos " x)/("sin " x)`
= cot x = RHS.
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