Calculus

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Solve identity,
(1-sin2x)/cos2x = cos2x/(1+sin2x)

I tried starting from the right side,
RS:
=(cos²x-sin²x)/(1+2sinxcosx)
=(cos²x-(1-cos²x))/(1+2sinxcosx)

and the right side just goes in circle. May I get a hint to start off?

  • Calculus -

    First of all since the angle is 2x throughout, let's just use y for 2x
    Secondly, you probably want to prove it as an identity, rather than solve it

    RS
    = cosy/(1+siny) [(1-siny)/(1-siny)]
    = cosy(1-siny)/(1- sin^2 y)
    = cosy(1-siny)/cos^2y
    = (1-siny)/cosy)
    = (1- sin 2x)/cos 2x
    = LS

  • Calculus -

    Cross multiply.

    cos^2(2x) = 1- sin^2(2x) = cos^2 2x
    q.e.d.

    x can be anything.

    That is why it is called an identity

  • Calculus -

    Got it, thanks!

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