Prove that: cot(^2) x- csc x/csc(^2) x = cos(^2) x- sin x

Question

Prove that: cot(^2) x- csc x/csc(^2) x
= cos(^2) x- sin x

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Everleigh 7 days 2021-11-23T18:14:26+00:00 1 Answer 0 views 0

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    2021-11-23T18:15:38+00:00

    \frac{cot^2x -cosec\ x}{cosec^2x} = cos^2x - sin\ x

    Solution:

    Given that we have to prove:

    \frac{cot^2x -cosec\ x}{cosec^2x} = cos^2x - sin\ x

    Take the L.H.S of above

    \frac{cot^2x -cosec\ x}{cosec^2x}

    Simplify

    \frac{cot^2x}{cosec^2x} -\frac{cosec\ x}{cosec^2x}\\\\Simplify\\\\\frac{cot^2x}{cosec^2x} -\frac{1}{cosec\ x}

    We know that,

    sin\ x = \frac{1}{cosec\ x}

    Therefore,

    \frac{cot^2x}{cosec^2x} -sin\ x

    Also we know that,

    cot^2x = \frac{cos^2\ x}{sin^2\ x}

    cosec^2\ x = \frac{1}{sin^2\ x}

    Thus we get,

    \frac{\frac{cos^2\ x}{sin^2 x}}{\frac{1}{sin^2\ x}} - sin\ x\\\\Simplify\\\\\frac{cos^2\ x}{sin^2 x} \times sin^2\ x - sin x\\\\Simplify\\\\cos^2\ x - sin\ x

    Thus,

    L.H.S = R.H.S

    Thus proved

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