Question

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

Answer 1

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

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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