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What is the exact answer to cos^2x - sin^2x = 0 for between 0 and 2pi?

I got x = pi/4 and x = 5pi/4. But the answer key says there are two additional answers which are 3pi/4 and 7pi/4. I don't know how to get these. Any help is much appreciated!

  • Math -

    cos^2x - sin^2x = 0
    cos (2x) = 0
    2x = π/2 or 2x = 3π/2

    x = π/4 or π = 3π/4

    but the period of cos (2x) is π
    so by adding π to any answer will yield a new answer
    new answers :
    π/4+π = 5π/4
    3π/4 + π = 7π/4

    Since I converted to cos 2x, there would be 2 complete cosine curves for 0 to 2π

    other way:

    cos^2 x = sin^2 x
    sin^2 x/cos^2 x = 1
    tan ^2 x = 1
    tan x = ± 1
    x = 45°, 135° , 225° , 315°
    or
    π/4 , 3π/4 , 5π/4 , 7π/4

  • Math -

    Thanks!

  • Math -

    There is an expression some people use that says, “What you put into it is what you get out of it.” People might use this expression to describe your skills at a sport or activity and how that relates to the amount of time and effort you spend practicing that activity. Does this expression apply to functions? How?

  • Math -

    There is an expression some people use that says, “What you put into it is what you get out of it.” People might use this expression to describe your skills at a sport or activity and how that relates to the amount of time and effort you spend practicing that activity. Does this expression apply to functions? How? Give an example to support your answer.

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