A) \[1-cos\text{ }A\text{ }cos\text{ }B\text{ }cos\text{ }C\]
B) \[1-2\text{ }cos\text{ }A\text{ }cos\text{ }B\text{ }cos\text{ }C\]
C) \[2\text{ }cos\text{ }A\text{ }cos\text{ }B\text{ }cos\text{ }C\]
D) \[1+cos\text{ }A\text{ }cos\text{ }B\text{ }cos\text{ }C\]
E) \[1+cos\text{ }A\text{ }2\text{ }cos\text{ }C\]
Correct Answer: B
Solution :
\[co{{s}^{2}}A+co{{s}^{2}}B+co{{s}^{2}}C\] \[=\frac{1}{2}[1+\cos 2A+1+\cos 2B+2{{\cos }^{2}}C]\] \[=\frac{1}{2}[2+2\cos (A+B)\cos (A-B)+2{{\cos }^{2}}C]\] \[=\frac{1}{2}[2+2\cos C\{\cos C-\cos (A-B)\}]\] \[=1-\cos C\{\cos (A+B)+\cos (A-B)\}\] \[=1-2\cos C\cos A\cos B\] \[=1-2\cos A\cos B\cos C\]
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