Solve the equation: ##\tan x ⋅\tan 4x = 1##

[chwala]

Homework Statement

Solve the equation $\tan x ⋅\tan 4x = 1$

Relevant Equations

trigonometry

I saw this link; is the approach here correct?

https://www.google.com/search?q=tan+x+tan+4x+=+1&oq=&gs_lcrp=EgZjaHJvbWUqCQgAECMYJxjqAjIJCAAQIxgnGOoCMgkIARAjGCcY6gIyCQgCECMYJxjqAjIJCAMQIxgnGOoCMgkIBBAjGCcY6gIyCQgFECMYJxjqAjIJCAYQIxgnGOoCMgkIBxAjGCcY6gLSAQkyNTc3ajBqMTWoAgiwAgE&sourceid=chrome&ie=UTF-8#fpstate=ive&vld=cid:7ceabfd6,vid:F4aNmm2QblY,st:0In my approach, i worked with:

...
$\cos 4x ⋅\cos x - \sin4 x⋅\sin x=0$

$\cos(4x + x)=0$

$\cos 5x = 0$

From here the solutions are determined easily...

cheers.

 

[pasmith]

This method has the advantage of not knowingly dividing by zero, which $$\tan (5x) = \frac{\tan x + \tan 4x}{1 - \tan x \tan 4x}$$ suffers from.

 

[bamboum]

I propose to write tan x= cot 4x because x cannot be 0. Then one will get:
3x=k Pi
where k integer positive or negative

 

[Mark44]

I propose to write tan x= cot 4x because x cannot be 0. Then one will get:
3x=k Pi
where k integer positive or negative

You have skipped a lot of steps going from $\tan(x) = \cot(4x)$ to $3x = k\pi$. How do you justify this large leap?

 

[bamboum]

Others solutions are related to x close to 0. Then tan x is x+1/3x^3. One obtains 4x^2+8/3x^4=1 giving x=0.218 or -0.218. If Taylor's serie is greatest in order perhaps one will get x close to 0.3