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 “ \displaystyle \text{1} ” in the numerator of the left-hand side with \displaystyle \text{sin}^{\text{2}}x+\text{cos}^{\text{2}}x\text{ } 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

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

The complete solution of the equation is

\displaystyle \left\{ \begin{array}{*{35}l} x=n\pi \\ x=\frac{3\pi }{4}+n\pi \\ \end{array} \right. ( \displaystyle n an arbitrary integer).