SOLUTION: If A is an angle such that tanA+secA=2, evaluate cosA.

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Question 1113584: If A is an angle such that tanA+secA=2, evaluate cosA.
Found 3 solutions by stanbon, ikleyn, MathTherapy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
If A is an angle such that tanA+secA=2, evaluate cosA.
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(sin/cos)+(1/cos) = 2
2*cos = (sin + 1)
cos(A) = (sin(A)+1)/2
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Cheers,
Stan H.
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Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
.
sin%28A%29%2Fcos%28A%29 + 1%2Fcos%28A%29 = 2


sin(A) + 1 = 2*cos(A)     =====>   Square both sides  ====>


sin%5E2%28A%29+%2B+2%2Asin%28A%29+%2B+1 = 4%2Acos%5E2%28A%29.


Introduce new variable t = sin(A). Then your last equation becomes


t%5E2+%2B+2t+%2B+1 = 4%2A%281-t%5E2%29


5t%5E2+%2B+2t+-+3 = 0


t%5B1%2C2%5D = %28-2+%2B-+sqrt%282%5E2+-+4%2A5%2A%28-3%29%29%29%2F%282%2A5%29 = %28-2+%2B-+8%29%2F10


t%5B1%5D = %28-2+%2B+8%29%2F10 = 6%2F10 = 3%2F5 = 0.6.


t%5B2%5D = %28-2+-+8%29%2F10 = -1.


Thus sin(A) = 0.6  or -1.  Then EITHER  cos(A) =  +/- sqrt%281-0.6%5E2%29 = +/- 0.8   OR  cos(A) = 0.


The original equation excludes  cos(A) = 0.  It leaves only two opportunities for cos(A):  It is EITHER  0.8  OR  -0.8.


Easy check with the original equation shows that only value cos(A) = 0.8 works.  Value cos(A) = -0.8 does not work.


Answer.  cos(A)  is   0.8.


The plot below CONFIRMS this answer:






Plot y = tan%28x%29+%2B+sec%28x%29 (red), y = 2 (green) and y = cos(x)

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For solving trigonometric equations, see the lessons
    - Solving simple problems on trigonometric equations
    - Solving typical problems on trigonometric equations
    - Solving more complicated problems on trigonometric equations
    - Solving advanced problems on trigonometric equations
    - OVERVIEW of lessons on calculating trig functions and solving trig equations
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Trigonometry: Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


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Writing by  @stanbon  is not the solution of the problem.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
If A is an angle such that tanA+secA=2, evaluate cosA.
matrix%281%2C3%2C+tan+%28A%29+%2B+sec+%28A%29%2C+%22=%22%2C+2%29

matrix%281%2C3%2C+tan+%28A%29%2C+%22=%22%2C+2+-+sec+%28A%29%29

matrix%281%2C3%2C+tan%5E2+%28A%29%2C+%22=%22%2C+4+-+4+sec+%28A%29+%2B+sec%5E2+%28A%29%29 -------- Squaring each side of equation

matrix%281%2C3%2C+sec%5E2+%28A%29+-+1%2C+%22=%22%2C+4+-+4+sec+%28A%29+%2B+sec%5E2+%28A%29%29 ----- Substituting matrix%281%2C3%2C+sec%5E2+%28A%29+-+1%2C+for%2C+tan%5E2+%28A%29%29

matrix%281%2C3%2C+sec%5E2+%28A%29+-+1+-+sec%5E2+%28A%29+%2B+4+sec+%28A%29%2C+%22=%22%2C+4%29

4 sec A = 5

matrix%281%2C3%2C+sec+%28A%29%2C+%22=%22%2C+5%2F4%29

highlight_green%28matrix%281%2C3%2C+cos+%28A%29%2C+%22=%22%2C+4%2F5%29%29