(1 + tan^2 theta) cot theta/ cosec^2 theta = tan theta​

Question

(1 + tan^2 theta) cot theta/
cosec^2 theta =
tan theta​

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Adalynn 4 weeks 2021-08-17T04:34:40+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-17T04:35:55+00:00

    To Prove :-

    • \sf \ \ \dfrac{(1+tan^2\theta)cot\theta}{cosec^2\theta}= tan\theta

    Identities Used :-

    • \underline{\boxed{\sf\ \ 1+tan^2\theta= sec^2\theta}}
    • \underline{\boxed{\sf\ \ sec\theta= \dfrac{1}{cos\theta}}}
    • \underline{\boxed{\sf\ \ Cosec\theta= \dfrac{1}{sin\theta}}}
    • \underline{\boxed{\sf\ \ tan\theta= \dfrac{sin\theta}{cos\theta}}}
    • \underline{\boxed{\sf\ \ cot\theta= \dfrac{1}{tan\theta}}}

    Proof :-

    \sf \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   :\implies LHS\\

     \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:\pink{ :\implies\sf \ \dfrac{\bigg(1+tan^2\theta\bigg)cot\theta}{cosec^2\theta}}\\

    \:  \:  \:  \:  \:  \:   \:  \:  \:  \:  \:  \:  \: \:    :\implies\sf \dfrac{\bigg(sec^2\theta\bigg)cot\theta}{cosec^2\theta}\\

    \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    :\implies\sf\ \dfrac{\bigg(\dfrac{1}{cos^2\theta}\bigg)cot\theta}{\dfrac{1}{sin^2\theta}}\\

    \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    :\implies\sf \ \dfrac{cot\theta}{cos^2\theta}\times sin^2\theta\\

     \:  \:  \:  \:  \:  \: \:  \:  \:  \:  \:  \:  \:   \:   :\implies\sf\ \dfrac{cot\theta\times sin^2\theta}{cos^2\theta}\\

    \:  \:  \:  \:   \:  \:  \:  \:  \:  \:  \: \:  \:  \:   :\implies\sf\ cot\theta\times \bigg(\dfrac{sin\theta}{cos\theta}\bigg)^2\\

     \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   :\implies\sf \ \dfrac{1}{\cancel{tan\theta}}\times \cancel{\bigg(tan^2\theta\bigg)}\\

    \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: \pink{   :\implies\sf {\boxed{\sf\ tan\theta}}}\\

    \sf\:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    :\implies RHS\\

    •  \sf Hence,\: Proved.. !!
    0
    2021-08-17T04:36:32+00:00

    Answer:

    (1+tan²)cot/cosec²

    sec²cot/cosec²

    sin²cot/cos²

    sin²cos/cos²sin

    sin/cos

    tan

    which is equal to rhs

    hence proved

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