SOLUTION: Given the equation: (sec x - 1)(sec x + 1) = 3 Solve for x: State which trigonometric identity/ies will give a result of 3. Please include a solution too, thank you so much.

Algebra ->  Trigonometry-basics -> SOLUTION: Given the equation: (sec x - 1)(sec x + 1) = 3 Solve for x: State which trigonometric identity/ies will give a result of 3. Please include a solution too, thank you so much.      Log On


   

Question 1173760: Given the equation:
(sec x - 1)(sec x + 1) = 3
Solve for x: State which trigonometric identity/ies will give a result of 3. Please include a solution too, thank you so much.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

%28sec%28x%29-1%29%28sec%28x%29%2B1%29+=+3

sec%5E2%28x%29-1+=+3 Difference of squares rule

tan%5E2%28x%29+=+3 Apply one of the pythagorean trig identities

tan%28x%29+=+%22%22%2B-sqrt%283%29 Apply the square root to both sides

tan%28x%29+=+sqrt%283%29 or tan%28x%29+=+-sqrt%283%29 Break up the plus minus into two separate equations

x+=+arctan%28sqrt%283%29%29 or x+=+arctan%28-sqrt%283%29%29 Arctangent is the same as inverse tangent.

x+=+pi%2F3+%2B+pi%2An or x+=+-pi%2F3%2Bpi%2An Use the unit circle. Note that tan(pi/3) = sqrt(3) and tan(-pi/3) = -sqrt(3). The pi%2An portion is needed to capture all possible solutions for each arctan equation solved.

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Answer:

The full solution set consists of x values such that
x+=+pi%2F3+%2B+pi%2An or x+=+-pi%2F3%2Bpi%2An
where n is any integer

One example solution is x+=+pi%2F3+%2B+2pi. In this case, we use n = 2 on the first equation shown above.