Question

If \(5\sin\theta=4\), then the value of \(\frac{\cosec\theta-\cot\theta}{\cosec\theta+\cot\theta}\) is

• A. \(-\frac{1}{4}\)
• B. \(-\frac{1}{2}\)
• C. \(\frac{1}{2}\)
• D. \(\frac{1}{4}\)

Hint:

When \(\sin\theta=\frac{4}{5}\), use the \(3-4-5\) triangle. Then \(\cos\theta=\frac{3}{5}\), \(\cosec\theta=\frac{5}{4}\), and \(\cot\theta=\frac{3}{4}\).

Answer 1

The Correct Option is D

We are given \[ 5\sin\theta=4. \] Therefore, \[ \sin\theta=\frac{4}{5}. \] Now, \[ \cosec\theta=\frac{1}{\sin\theta}. \] So, \[ \cosec\theta=\frac{5}{4}. \] Using the identity \[ \sin^2\theta+\cos^2\theta=1, \] we get \[ \left(\frac{4}{5}\right)^2+\cos^2\theta=1. \] \[ \frac{16}{25}+\cos^2\theta=1. \] \[ \cos^2\theta=\frac{9}{25}. \] Taking positive value, \[ \cos\theta=\frac{3}{5}. \] Now, \[ \cot\theta=\frac{\cos\theta}{\sin\theta}. \] \[ \cot\theta=\frac{\frac{3}{5}}{\frac{4}{5}}. \] \[ \cot\theta=\frac{3}{4}. \] Now substitute in the expression: \[ \frac{\cosec\theta-\cot\theta}{\cosec\theta+\cot\theta} = \frac{\frac{5}{4}-\frac{3}{4}}{\frac{5}{4}+\frac{3}{4}}. \] \[ = \frac{\frac{2}{4}}{\frac{8}{4}}. \] \[ = \frac{2}{8}. \] \[ = \frac{1}{4}. \] Hence, the required value is \[ \frac{1}{4}. \]