Question:
Find the value of cosecθ (1 − cosθ) (cosecθ + cotθ).
cosecθ (1 − cosθ) (cosecθ + cotθ) का मान ज्ञात कीजिए।
Options:
- a) 2
- b) -1
- c) 1
- d) 0
- a) 2
- b) -1
- c) 1
- d) 0
correct answer : c)
Explanation:
We can write the expression in terms of $\sin\theta$ and $\cos\theta$:
$\csc\theta (1 – \cos\theta) (\csc\theta + \cot\theta)$
$= \frac{1 – \cos\theta}{\sin\theta} \times \left(\frac{1}{\sin\theta} + \frac{\cos\theta}{\sin\theta}\right)$
$= \frac{1 – \cos\theta}{\sin\theta} \times \frac{1 + \cos\theta}{\sin\theta}$
$= \frac{(1 – \cos\theta)(1 + \cos\theta)}{\sin^2\theta}$
$= \frac{1 – \cos^2\theta}{\sin^2\theta}$
Since $1 – \cos^2\theta = \sin^2\theta$:
$= \frac{\sin^2\theta}{\sin^2\theta} = 1$.
Hence, option (c) is the correct answer.
हम व्यंजक को $\sin\theta$ और $\cos\theta$ के रूप में लिख सकते हैं:
$\csc\theta (1 – \cos\theta) (\csc\theta + \cot\theta)$
$= \frac{1 – \cos\theta}{\sin\theta} \times \left(\frac{1}{\sin\theta} + \frac{\cos\theta}{\sin\theta}\right)$
$= \frac{1 – \cos\theta}{\sin\theta} \times \frac{1 + \cos\theta}{\sin\theta}$
$= \frac{(1 – \cos\theta)(1 + \cos\theta)}{\sin^2\theta}$
$= \frac{1 – \cos^2\theta}{\sin^2\theta}$
चूंकि $1 – \cos^2\theta = \sin^2\theta$:
$= \frac{\sin^2\theta}{\sin^2\theta} = 1$।
अतः, विकल्प (c) सही उत्तर है।