Question

Simplify tan9x-tan5x / 1+tan9xtan5x
(SHOW WORK)

Answer 1

The simplest form is tan(4x)

Explanation:

* Lets revise the identity of the compound angles

-
`tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))`

-
`tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b))`

* Lets solve the problem

- Let 9x = 5x + 4x

∴ tan(9x) = tan(5x + 4x)

- Use the rule of the compound angle

∵
`(tan(9x)-tan(5x))/(1+tan(9x)tan(5x))` ⇒ (1)

∵
`tan(5x+4x)=(tan(5x)+tan(4x))/(1-tan(5x)tan(4x))` ⇒ (2)

∵ tan(9x) = equation (2)

- Substitute (2) in (1)

∴
`((tan(5x)+tan(4x))/(1-tan(5x)tan(4x))-tan(5x))/(1+((tan(5x)+tan(4x))/(1-tan(5x)tan(4x)))tan(5x))`

- Multiply up and down by (1 - tan(5x)tan(4x))

∴
`(tan(5x)+tan(4x)-tan(5x)[1-tan(5x)tan(4x)])/(1-tan(5x)tan(4x)+tan(5x)[tan(5x)+tan(4x)])`

- Simplify up and down

∴
`(tan(5x)+tan(4x)-tan(5x)+tan^(2)(5x)tan(4x))/(1-tan(5x)tan(4x)+tan^(2)(5x)+tan(5x)tan(4x) )`

∴
`(tan(4x)+tan^(2)(5x)tan(4x))/([1+tan^(2)(5x)])`

- Take tan(4x) as a common factor up

∴
`(tan(4x)[1+tan^(2)(5x)])/([1+tan^(2)(5x)])`

- Cancel [1 + tan²(5x)] up and down

∴ The answer is tan(4x)