Question : If $m = \sec \theta- \tan \theta$ and $n = \operatorname{cosec} \theta + \cot \theta$, then what is the value of $m + n(m-1)$?
Option 1: 2
Option 2: 1
Option 3: 0
Option 4: –1
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Correct Answer: –1
Solution : Given: $m = \sec \theta- \tan \theta$ and $n = \operatorname{cosec} \theta + \cot \theta$ $⇒m=\frac{(1 - \text{sin $\theta$})}{\text{cos $\theta$}}$ and $⇒n=\frac{(1 + \text{cos $\theta$})}{\text{sin $\theta$}}$ $m + n(m - 1)$ $= m + nm - n$ $=\frac{(1 - \text{sin $\theta$})}{\text{cos $\theta$}}+\frac{(1 - \text{sin $\theta$})}{\text{cos $\theta$}}×\frac{(1 + \text{cos $\theta$})}{\text{sin $\theta$}}-\frac{(1 + \text{cos $\theta$})}{\text{sin $\theta$}}$ $=\frac{(\sin\theta-\sin^2\theta+1-\sin\theta\cos\theta+\cos\theta-\sin\theta-\cos\theta-\cos^2\theta)}{\text{sin $\theta$} × \text{cos $\theta$}}$ $=\frac{-(\sin^2\theta+\cos^2\theta)+1-\sin\theta\cos\theta}{\text{sin $\theta$} × \text{cos $\theta$}}$ $=\frac{- \text{sin $\theta$} × \text{cos $\theta$}}{\text{sin $\theta$}× \text{cos $\theta$}}$ $= - 1$ Hence, the correct answer is –1.
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Question : What is the value of $\frac{\cot \theta+\operatorname{cosec} \theta-1}{\cot \theta-\operatorname{cosec} \theta+1}$?
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