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Solving Simultaneous Trigonometric Equations

If x=θ+1 and ...

Question

If $x = s e c θ + 1 a n d y = t a n θ - 1$ , then $(x - 1)^{2} - (y + 1)^{2}$ is equal to
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Solution

$x = s e c θ + 1 \Rightarrow (x - 1) = s e c θ$
$y = t a n θ - 1 \Rightarrow y + 1 = t a n θ$
$\Rightarrow (x - 1)^{2} - (y + 1)^{2} = s e c^{2} θ - t a n^{2} θ = 1$

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