Step-by-step explanation:
tan(x)sec(x)-2tan(x)=0
Solving these equations usually starts with using algebra and/or Trig. identities to transform the equation so that you have one or more equations of the form:
trigfunction(expression) = number
With this equation the transformation is not too difficult. First we factor out tan(x):
tan(x)(sec(x)-2)=0
And then use the Zero Product Property:
tan(x) = 0 or sec(x)-2 = 0
Adding 2 to each side of the second equation we get:
tan(x) = 0 or sec(x) = 2
We now have two equations of the desired form.
The next step is to write the general solution for each equation. The general solution expresses all the solutions to the equations. We'll start with:
tan(x) = 0
We should recognize that 0 is a special angle value for tan. So we will not need our calculators. tan is zero angles of 0, pi and for all other angles that are co-terminal with these two. We express this with:
x+=+0+%2B+2pi%2An
x+=+pi+%2B+2pi%2An
The n's in these equations represent any integer. By replacing "n" with an integer you end up with an x that is 0, pi or some other angle that is co-terminal.
Next
ย
sec(x) = 2
You may not recognize that 2 is a special angle value for sec. Since cos is the reciprocal of sec, an angle whose sec is 2 will have a cos of 1/2:
cos%28x%29+=+1%2F2
We should recognize that 1/2 is definitely a special angle value for cos. The reference angle is pi%2F3. And since the 1/2 is positive and cos (and sec) are positive in the 1st and 4th quadrants, the angles we are looking for will terminate in either the 1st or 4th quadrant and have a reference angle of pi%2F3:
x+=+pi%2F3+%2B+2pi%2An
x+=+-pi%2F3+%2B+2pi%2An (or x+=+5pi%2F3+%2B+2pi%2An)
Altogether the general solution to your equation is:
x+=+0+%2B+2pi%2An
x+=+pi+%2B+2pi%2An
x+=+pi%2F3+%2B+2pi%2An
x+=+-pi%2F3+%2B+2pi%2An (or x+=+5pi%2F3+%2B+2pi%2An)
Many, but not all, of these problems ask you to find a specific solution. For example: "Find the least positive solution to ..." or "Find all solutions to ... that are between zero and 2pi". When specific solutions are requested you use the general solution equation(s) and various integer values for "n" until you have found the requested solution(s).
In this case the problem does not ask for a specific solution. It asks for "all real values". So the general solution is the solution when no specific solution is requested.
P.S. The period for tan (and cot) is just pi. So instead of 2 general solution equations for tan (or cot) equations, you can just use one. In this case
x+=+0+%2B+2pi%2An
x+=+pi+%2B+2pi%2An
can be replaced with just:
x+=+0+%2B+pi%2An
