Solution 4.4:8c

From Förberedande kurs i matematik 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 sin2x+cos2x using the Pythagorean identity. This means that the equation's left-hand side can be written as

1cos2x=cos2xcos2x+sin2x=1+sin2xcos2x=1+tan2x

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

1+tan2x=1−tanx

If we substitute t=tanx , we see that we have a second-degree equation in t , which, after simplifying, becomes t2 +t=0 and has roots t=0 and t=−1. There are therefore two possible values for tanx, tanx=0 tan x =0 or tanx=−1 The first equality is satisfied when x=n for all integers n, and the second when x=43+n.

The complete solution of the equation is

x=nx=43+n  ( \displaystyle n an arbitrary integer).