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Question
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
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Solution
\[\text{ LHS }\hspace{0.167em} = 2\cot(\alpha - \beta)\]
\[ = \frac{2(1 + \tan\alpha\tan\beta)}{\left[ \tan\alpha - \tan\beta \right]}\]
\[= \frac{2 + 2\left( x + 1 \right)\left( x - 1 \right)}{\left( x + 1 - x + 1 \right)}\]
\[ = \frac{2 + 2 x^2 - 2}{2}\]
\[ = \frac{2 x^2}{2}\]
\[ = x^2 \]
= RHS
Hence proved.
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