Question

$\colorbox{blue}{ \colorbox{red} { \colorbox{maroon}{\color{pink} \text{find \: \( \color{pink} \tt\dfrac{dy}{dx}:\)- } \text{\( \color{pink} \tt \: x=a \sec ^{3} \theta, y=a \tan ^{3} \theta \)}}}}$
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Answer 1

Answer:

Step-by-step explanation:

We have,

$\rm{x=a\,sec^3(\theta)\,\,\,\,\,\&\,\,\,\,\,y=a\,tan^3(\theta)}$

$\rm{\implies\,\dfrac{dx}{d\theta}=3a\,sec^2(\theta)\cdot\,sec(\theta)\,tan(\theta)\,\,\,\,\,\&\,\,\,\,\,\dfrac{dy}{d\theta}=3a\,tan^2(\theta)\cdot\,sec^2(\theta)}$

Now,

$\rm{\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{d\theta}}{\dfrac{dx}{d\theta}}=\dfrac{3a\,tan^2(\theta)\,sec^2(\theta)}{3a\,sec^2(\theta)\cdot\,sec(\theta)\,tan(\theta)}}$

$\rm{\implies\,\dfrac{dy}{dx}=\dfrac{tan(\theta)}{sec(\theta)}}$

$\rm{\implies\,\dfrac{dy}{dx}=sin(\theta)}$

Answer 2

Answer:

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