Question : If $(\frac{\sec \theta-1}{\sec \theta+1})^n=\operatorname{cosec} \theta-\cot \theta$, then $n=?$
Option 1: 1
Option 2: 0.5
Option 3: –1
Option 4: –0.5
Correct Answer: 0.5
Solution :
Given expression,
$(\frac{\sec \theta-1}{\sec \theta+1})^n=\operatorname{cosec} \theta-\cot \theta$
Considering L.H.S.,
We know, $\sec\theta=\frac{1}{\cos\theta}$
= $\left\{\frac{\frac{1}{\cos\theta}-1}{\frac{1}{\cos\theta}+1}\right\}^n$
= $\left\{\frac{1-\cos\theta}{1+\cos\theta}\right\}^n$
Multiplying and dividing by $1-\cos\theta$, we get,
= $\left\{\frac{1-\cos\theta}{1+\cos\theta}\times\frac{1-\cos\theta}{1-\cos\theta}\right\}^n$
= $\left\{\frac{(1-\cos\theta)^2}{1-\cos^2\theta}\right\}^n$
= $\left\{\frac{(1-\cos\theta)^2}{\sin^2\theta}\right\}^n$
Now consider R.H.S,
$\operatorname{cosec} \theta-\cot \theta$
We know, $\operatorname{cosec} \theta=\frac{1}{\sin\theta}$ and $\cot\theta=\frac{\cos\theta}{\sin\theta}$
= $\frac{1}{\sin\theta}-\frac{\cos\theta}{\sin\theta}$
= $\frac{1-\cos\theta}{\sin\theta}$
Equating with L.H.S, we get,
⇒ $2n=1$
$\therefore n=0.5$
Hence, the correct answer is 0.5.
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