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:

If \(\sin\theta=\frac{4}{5}\), use the \(3-4-5\) triangle to quickly find \(\cos\theta=\frac{3}{5}\).

Answer 1

The Correct Option is D

Concept:
Use the given value of \(\sin\theta\), then find \(\cosec\theta\) and \(\cot\theta\).

Step 1: Given: \[ 5\sin\theta=4 \] \[ \sin\theta=\frac{4}{5} \]

Step 2: Therefore: \[ \cosec\theta=\frac{1}{\sin\theta}=\frac{5}{4} \]

Step 3: Using the \(3-4-5\) triangle: \[ \sin\theta=\frac{4}{5} \] So, \[ \cos\theta=\frac{3}{5} \]

Step 4: Now: \[ \cot\theta=\frac{\cos\theta}{\sin\theta} \] \[ \cot\theta=\frac{\frac{3}{5}}{\frac{4}{5}} \] \[ \cot\theta=\frac{3}{4} \]

Step 5: Substitute in the expression: \[ \frac{\cosec\theta-\cot\theta}{\cosec\theta+\cot\theta} = \frac{\frac{5}{4}-\frac{3}{4}}{\frac{5}{4}+\frac{3}{4}} \] \[ = \frac{\frac{2}{4}}{\frac{8}{4}} \] \[ = \frac{2}{8} \] \[ = \frac{1}{4} \] \[ \boxed{\frac{1}{4}} \]