If \(f(t)=\frac{1+\operatorname{cosec} t}{1-\operatorname{cosec} t}\) for \(0 < t < \frac{\pi}{2}\) and…
If \(f(t)=\frac{1+\operatorname{cosec} t}{1-\operatorname{cosec} t}\) for \(0 < t < \frac{\pi}{2}\) and \(f^{\prime}(t)=f(t) g(t)\), then \(g(t)=\)
- \(-4 \operatorname{cosec} 2 t\)
- \(4 \operatorname{cosec} 2 t\)
- \(2 \sin 2 t\)
- \(4 \operatorname{cosec} t\)
Solution
Given, \(f(t)=\frac{1+\operatorname{cosec} t}{1-\operatorname{cosec} t}\)
Differentiating w.r.t. \(t\), we get
\((1-\operatorname{cosec} t)(-\operatorname{cosec} t \cot t)\)
\(\begin{aligned}
f^{\prime}(t) & =\frac{-(1-\operatorname{cosec} t)(\operatorname{cosec} t \cot t)}{(1-\operatorname{cosec} t)^2} \\
& =\frac{-\operatorname{cosec} t \cot t+\operatorname{cosec}^2 t \cot t}{(1-\operatorname{cosec} t)^2} \\
& =\frac{-2 \operatorname{cosec} t \cot t}{(1-\operatorname{cosec} t)^2} \times \frac{1+\operatorname{cosec} t}{1+\operatorname{cosec} t} \\
& =\left(\frac{1+\operatorname{cosec} t}{1-\operatorname{cosec} t}\right)\left(-\frac{2 \operatorname{cosec} t \cot t}{1-\operatorname{cosec}^2 t}\right) \\
& =f(t) \frac{2 \operatorname{cosec} t \cot t}{\cot ^2 t}=f(t) \frac{2 \operatorname{cosec}^2 t}{\cot ^2 t} \\
& =f(t) \frac{2}{\sin t \cos t}=f(t)(4 \operatorname{cosec} 2 t)=f(t) g(t) \quad \text{[given]}
\end{aligned}\)
So, \(g(t)=4 \operatorname{cosec} 2 t\)
Asked in: AP EAMCET 2019 (22 Apr Shift 1)
Practice more Differentiation questions on Aicharya