If - +1=0, then prove that =-1. - Vidyadip

Question

If $\sin \alpha \sin \beta-\cos \alpha \cos \beta+1=0$, then prove that $\cot \alpha \tan \beta=-1$.

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Answer

$\begin{aligned}
& \sin \alpha \sin \beta-\cos \alpha \cos \beta+1=0 \\
& \therefore \cos \alpha \cos \beta-\sin \alpha \sin \beta=1 \\
& \therefore \cos (\alpha+\beta)=1 \\
& \therefore \alpha+\beta=0 \ldots \ldots[\because \cos 0=1] \\
& \therefore \beta=-\alpha \\
& \text { L.H.S. }=\cot \alpha \tan \beta \\
& =\cot \alpha \tan (-\alpha) \\
& =-\cot \alpha \tan \alpha \\
& =-1 \\
& =\text { R.H.S. }
\end{aligned}$

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