Question

In any triangle ABC, if cos A cos B + sin Asin B sin C=1 then prove that triangle is an isosceles right angled.

JEE/Engineering Exams

Maths

Solution

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we have ( cos A cos B+sin A sin B sin C=1 ) ( Rightarrow cos A cos B+sin A sin B-sin A sin B ) ( +sin A sin B sin C=1 ) ( Rightarrow cos (A-B)-sin A sin B(1-sin C)=0 ) ( Rightarrow 2 sin ^{2}left(frac{A-B}{2}right)+sin A sin B(1-sin C)=0 ) In this term This teem is squase ( A, B, C ) all are term : it is less than ( 180^{circ} ) but positine ( therefore ) Sin ( A ) sinB and ( sin C ) are the and less than 1 . So second term is also positine But RHS = 0 . Which in possible oney if each tesm is zero on UHS. ( therefore 2 sin ^{2}left(frac{A-B}{2}right)=0 ) men ( A=B ) and when ( sin A sin B(1-sin C)=0 ) ( 0