If in a triangle ABC #cosAcosB+sinAsinBsinC=1# then how will you prove that the triangle is right angled and isosceles?
Please see below.
Multiplying both sides by
Note that all three terms contained above are positive, as while first two terms are squares and hence positive, third term is positive as sine of angles
But, their sum is zero and hence each term is equal to zero, i.e.
Hence the triangle is isosceles and right angled.
Now in above relation the first term being squared quantity will be positive.In the second term A,B and C all are less than
So sinA ,sinB and sinC all are positive and less than 1.So the 2nd term as a whole is positive.
It is only possible iff each term becomes zero.
and when 2nd term=0 then
0< A and B <180
So in triangle ABC