verify ((cot(theta)-tan(theta))/((cot(theta)-tan(theta))=cos(2 theta)?

only got as far as

((1/tan(theta)-tan(theta))/((1/tan(theta)-tan(theta))=cos(2 theta)

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  • Anonymous
    6 years ago
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    [cotΘ - tanΘ] / [cotΘ - tanΘ] = cos2Θ ?

    This would equal to 1 since both numerator and denominator are same. If it's:

    [cotΘ - tanΘ] / [cotΘ + tanΘ] = cos2Θ then:

    LHS ----> [(1/tanΘ) - tanΘ] / [(1/tanΘ) + tanΘ] ---> Convert tanΘ to sinΘ and cosΘ

    [(cosΘ/sinΘ) - (sinΘ/cosΘ)] / [(cosΘ/sinΘ) + (sinΘ/cosΘ)]

    [(cos²Θ - sin²Θ) / sinΘcosΘ] / [(cos²Θ + sin²Θ) / sinΘcosΘ]

    Both denominators cancel each other and you're left with:

    [cos²Θ - sin²Θ] / [cos²Θ + sin²Θ] ----> cos²Θ + sin²Θ = 1

    Now you have ---> cos²Θ - sin²Θ which is equal to cos2Θ

    Hence ---> cos²Θ - sin²Θ = cos2Θ. --> Identity proven.

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