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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