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  4. If Δ ABC, if cos A cos B+ sin A sin B sin C=1 and C=(π/2), then A: B=

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Q. If $\Delta \,ABC$, if $\cos\, A \cos\, B+\sin \,A \sin\, B \sin \,C=1$ and $C=\frac{\pi}{2}$, then $A: B=$

TS EAMCET 2018

Solution:

Given,
$\cos \,A \,\cos \,B+\sin \,A \sin \,B \sin C=1$ and $C=\frac{\pi}{2}$
$\Rightarrow \, \sin C=\sin \frac{\pi}{2}=1$
So, $\cos \,A \cos\, B+\sin A \sin B(1)=1$
$\Rightarrow \, \cos (A-B)=1$
$[\because \cos (A+B)-\cos A \cos B-\sin A \sin B]$
$\Rightarrow \, 1-\cos (A-B)=0$
$\Rightarrow \, 2 \sin ^{2}\left(\frac{A-B}{2}\right)=0$
It is only possible, when
$A-B=0$
$\Rightarrow \, A=B$
$\Rightarrow \, \frac{A}{B}=1$
Hence, $A : B=1 : 1$