Question

If \( \cosec\theta + \cot\theta = 5 \), then \( \sin\theta = \)

• A. \( \dfrac{1}{5} \)
• B. \( \dfrac{5}{26} \)
• C. \( \dfrac{5}{13} \)
• D. \( \dfrac{1}{13} \)

Hint:

Whenever expressions involve \( \cosec\theta \) and \( \cot\theta \), multiplying by their conjugate simplifies calculations.

Answer 1

The Correct Option is C

Step 1: Use the identity.
\[ (\cosec\theta + \cot\theta)(\cosec\theta - \cot\theta) = 1 \] Step 2: Find \( \cosec\theta - \cot\theta \).
\[ \cosec\theta - \cot\theta = \frac{1}{5} \] Step 3: Solve the system.
\[ \cosec\theta + \cot\theta = 5 \] \[ \cosec\theta - \cot\theta = \frac{1}{5} \] Adding, \[ 2\cosec\theta = \frac{26}{5} \Rightarrow \cosec\theta = \frac{13}{5} \] Step 4: Find \( \sin\theta \).
\[ \sin\theta = \frac{5}{13} \] Step 5: Conclusion.
The value of \( \sin\theta \) is \( \dfrac{5}{13} \).