If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1 - Vidyadip

Question

If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1

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Answer

$
\begin{aligned}
& \text { LHS }=m n=(\sec \theta+\tan \theta)(\sec \theta-\tan \theta) \\
& \Rightarrow \text { LHS }=\sec ^2 \theta-\tan ^2 \theta \quad\left[\text { Because }(a-b)(a+b)=a^2-b^2\right] \\
& \Rightarrow \text { LHS }=1\left[\text { Since } 1+\tan ^2 \theta=\sec ^2 \theta\right]
\end{aligned}
$
Hence, $m n=1$

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