How can I check if the period of a trigonometric equation solution is npi" or "2npi ?

(sinx+cosx)*sqrt(2)=tanx+cotx
Solving this equation I got x= pi/4+npi , but my professor said the solution should be x= pi/4+2npi
How is this checked?

1 Answer
Jul 8, 2018

See the explanation, and graph for verification.

Explanation:

The least positive P for which f (x + P) = f (x ) is the period of f( x ).

The period for both sin x and cos x is 2pi.

The period for both tan x and cot x is pi.

The LCM is 2pi .

So, this is the overall period for

f( x ) = sqrt2 ( cos x + sin x ) - ( tan x + cot x).

You can verify that

f( x + 2pi ) = f( x ) but

f( x + pi )= - sqrt2 ( cos x + sin x ) - ( tan x + cot x) ne f(x).

So, the general solution is

x = pi/4 + 2npi, n = 0, +-1, +-2, +-3, ..

Graph reveals this.
graph{y - sqrt 2 (sin x + cos x )+tan x + cot x = 0[-20 20 -10 10]}

The graph braces x-axis, for the solutions, at

x = ... 18.6, -11.8, -5.5, 0.79, 7.07, 13.4 ... for

= ...- (23/4)pi, -(15/4)pi, -(7/4)pi, pi/4, (9/4)pi, (17/4)pi...,

respectively, as approximations.