Question
If the $$\vartriangle ABC$$ is acute angled at $$C$$ then
A.
$$\cos 2A + \cos 2B - \cos 2C < 1$$
B.
$$\cos 2A + \cos 2B + \cos 2C > 1$$
C.
$${\cos ^2}A + {\cos ^2}B + {\cos ^2}C > 1$$
D.
None of these
Answer :
$$\cos 2A + \cos 2B - \cos 2C < 1$$
Solution :
$$\eqalign{
& {\text{In }}\vartriangle ABC,\cos 2A + \cos 2B - \cos 2C = 1 - 4\sin A\sin B\cos C < 1\,\,{\text{for all }}A,B. \cr
& \cos 2A + \cos 2B + \cos 2C = - 1 - 4\cos A\cos B\cos C < - 1\,\,{\text{for acute }}A,B. \cr
& {\cos ^2}A + {\cos ^2}B + {\cos ^2}C = 1 - 2\cos A\cos B\cos C < 1\,\,{\text{for acute }}A,B. \cr} $$