Aus Online Mathematik Brückenkurs 1

When we have a trigonometric equation which contains a mixture of different trigonometric functions, a useful strategy can be to rewrite the equation so that it is expressed in terms of just one of the functions. Sometimes, it is not easy to find a way to rewrite it, but in the present case a plausible way is to replace the “1” in the numerator of the left-hand side with \displaystyle \sin^2\!x + \cos^2\!x using the Pythagorean identity. This means that the equation's left-hand side can be written as

\displaystyle \frac{1}{\cos ^{2}x} = \frac{\cos^2\!x + \sin^2\!x}{\cos^2\!x} = 1 + \frac{\sin^2\!x}{\cos^2\!x} = 1+\tan^2\!x

and the expression is then completely expressed in terms of tan x,

\displaystyle 1 + \tan^2\!x = 1 - \tan x\,\textrm{.}

If we substitute \displaystyle t=\tan x, we see that we have a quadratic equation in t, which, after simplifying, becomes \displaystyle t^2+t=0 and has roots \displaystyle t=0 and \displaystyle t=-1. There are therefore two possible values for \displaystyle \tan x, \displaystyle \tan x=0 or \displaystyle \tan x=-1\,. The first equality is satisfied when \displaystyle x=n\pi for all integers n, and the second when \displaystyle x=3\pi/4+n\pi\,.

The complete solution of the equation is

\displaystyle \left\{\begin{align} x &= n\pi\,,\\[5pt] x &= \frac{3\pi}{4}+n\pi\,, \end{align}\right.

where n is an arbitrary integer.