If \(\sec\theta+\text{cosec}\theta = a\) and \(\tan\theta+\cot\theta = b\), find the value of \(\text{cosec}\,2\theta\).
\(\dfrac{b}{2}\)
\(\tan\theta+\cot\theta = \dfrac{\sin\theta}{\cos\theta}+\dfrac{\cos\theta}{\sin\theta} = \dfrac{1}{\sin\theta\cos\theta} = \dfrac{2}{\sin2\theta} = b\).
So \(\sin2\theta = \dfrac{2}{b}\).
\(\text{cosec}\,2\theta = \dfrac{1}{\sin2\theta} = \dfrac{b}{2}\).
Hence, the value of \(\text{cosec}\,2\theta\) is \(\tfrac{b}{2}\).
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