Math Identities qsts, will award best answer, pls help?

prove the following: 

1.   (cosθ/ secθ -1) + (cosθ/ secθ +1) = 2cot^2*θ

2.   (sinθ + tanθ) / (cosθ +1) = tanθ

3.   (1-cos 2θ) / (sin 2θ) = tanθ

Answering even one question would be helpful, thanks so much!

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  • 6 months ago
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    1) [(cosθ/(secθ - 1)] + [(cosθ/(secθ + 1)] = 2cot²θ

    Considering the left side we have:

    cosθ/(secθ - 1) + cosθ/(secθ + 1)

    so, [cosθ(secθ + 1) + cosθ(secθ - 1)]/(secθ - 1)(secθ + 1)

    => [2cosθsecθ]/(sec²θ - 1)

    Now, sec²θ => tan²θ + 1 so,

    [2cosθsecθ]/tan²θ

    Also, secθ => 1/cosθ so,

    [2cosθ/cosθ]/tan²θ

    i.e. 2/tan²θ

    => 2cot²θ...RHS

    :)>

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