Q67 / 160 · TS EAMCET 2005 · Mathematics

$A+B=C \quad \Rightarrow$ $\cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C$ is equal to

  • Correct answer
    A
    1
  • B
    2
  • C
    0
  • D
    3

Solution

Given that, $A+B=C$ $\begin{aligned} & \text { Now, } \cos ^2 A+\cos ^2 B+\cos ^2 C \\ & =\frac{1+\cos 2 A}{2}+\frac{1+\cos 2 B}{2}+\cos ^2 C \\ & =1+\frac{1}{2}(\cos 2 A+\cos 2 B)+\cos ^2 C \\ & =1+\frac{2}{2}[\cos (A+B) \cos (A-B)]+\cos ^2 C \\ & =1+\cos C[\cos (A-B)+\cos (A+B)] \\ & =1+2 \cos C \cos B \cos A \\ & \Rightarrow \cos ^2 A+\cos ^2 B+\cos ^2 C-2 \cos A \cos B \cos C=1 \end{aligned}$

Hence, the correct answer is option: 'A'.

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