If cosectheta + cottheta = 4/3. Find cosectheta, cottheta, sintheta, tantheta.​

Question

If cosectheta + cottheta = 4/3. Find cosectheta, cottheta, sintheta, tantheta.​

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Natalia 1 month 2021-08-17T09:54:40+00:00 1 Answer 0 views 0

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    2021-08-17T09:56:30+00:00

    \huge\boxed{\underline{\underline{\green{\tt Solution}}}}

     \displaystyle \: cosec \theta + cot \theta =  \frac{4}{3}  \: ..........(1)

    We know that

     \displaystyle \:  {cosec}^{2}  \theta -  {cot}^{2}  \theta = 1

     \implies \: \displaystyle \: (cosec \theta + cot \theta )(cosec \theta  -  cot \theta ) = 1

     \implies \: \displaystyle \:  \frac{4}{3}  \times (cosec \theta  -  cot \theta ) = 1

     \implies \: \displaystyle \:  (cosec \theta  -  cot \theta ) =  \frac{3}{4}  \: .......(2)

    Equation (1)+(2) gives

     \implies \: \displaystyle \:  2cosec \theta   =  \frac{4}{3}  +  \frac{3}{4}

     \implies \: \displaystyle \:  2cosec \theta   =  \frac{25}{12}

     \implies \: \displaystyle \:  cosec \theta   =  \frac{25}{24}

    So

     \displaystyle \:  sin \:\theta  =  \frac{1}{cosec \:\theta}  =  \frac{24}{25}

    From(1)

    \implies \: \displaystyle \:  cot \theta   =   \frac{4}{3}  - \frac{25}{24}   =  \frac{7}{24}

    \implies \: \displaystyle \:  tan \theta   =   \frac{1}{cot\theta  }  = \frac{24}{7}

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