1. ## tan(a*x)=?

hello
does anyone knows that is there any sequence for tan(a*x) ?
for example for tan(2*x)=2*tan(x)/(1-tan^2(x))
and tan 3x = 3tanx - (tan x)^3]/[1 - 3(tanx)^2]

2. here you are: (follwing is a link)

prouver que tan(nx) = ... (forum ilemaths)

Bernard

3. Hello, abdolah!

Does anyone knows that is there any sequence for tan(nx) ?

for example: .tan(2x) .= .2*tan(x) / [1 - (tan x)^2]

and: tan 3x .= .[3*tanx - (tan x)^3] / [1 - 3(tan x)^2]

Back in college, I discovered a delightful pattern for this.

Example: tan(5x).

Consider the expansion of (1 + tan x)^5:

. . 1 + 5(tan x) + 10(tan x)^2 + 10(tan x)^3 + 5(tan x)^4 + (tan x)^5

Write alternate terms in the denominator and numerator of a fraction.

. . . . 5(tan x) . . . . 10(tan x)^3 . . . . (tan x)^5
. .--------------------------------------------------------
. . 1 . . . . 10(tan x)^2 . . . . 5(tan x)^4

Insert alternating signs in the numerator and in the denominator:

. . . .+5(tan x) . . . . - 10(tan x)^3 . . . . + (tan x)^5
. .--------------------------------------------------------
. .+1 . . . . - 10(tan x)^2 . . . . +5(tan x)^4

. . . . . . . . . . . . . . . . .5(tan x) - 10(tan x)^3 + (tan x)^5
Therefore: .tan(5x) .= .-----------------------------------------
. . . . . . . . . . . . . . . . . . 1 - 10(tan x)^2 + 5(tan x)^4

4. This technique can be used for $\sin(nx)$ and $\cos(nx)$

Example: . $\sin(5x)$ and $\cos(5x)$

Consider the expansion of: .(cos x + sin x)^5

. . (cos x)^5 + 5(cos x)^4(sin x) + 10(cos x)^3(sin x)^2

. . . . . . + 10(cos x)^2(sin x)^3 + 5(cos x)(sin x)^4 + (sin x)^5

. . . . . . 5(cos x)^4(sin x) . . . . 10(cos x)^2(sin x)^3 . . . . (sin x)^5

. . (cos x)^5 . . . . 10(cos x)^3(sin x)^2. . . . 5(cos x)(sin x)^4

Insert alternating signs in the first row and in ths second row:

. . . . . .+5(cos x)^4(sin x) . . . .-10(cos x)^2(sin x)^3 . . . .+(sin x)^5

. .+(cos x)^5 . . . .-10(cos x)^3(sin x)^2. . . .+5(cos x)(sin x)^4

And we have:

. .sin(5x) .= .5(cos x)^4(sin x) - 10(cos x)^2(sin x)^3 + (sin x)^5

. .cos(5x) .= .(cos x)^5 - 10(cos x)^3(sin x)^2 + 5(cos x)(sin x)^4

5. thanks , very good answers

,

,

### expand the tan(nx) in the term of tanx

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