Question

In triangle $A B C$, if $\begin{vmatrix}1 & a & b \\ 1 & c & a \\ 1 & b & c\end{vmatrix}=0$ then the value of $\sin ^2 A+\cos ^2 B+\tan ^2 C$ is equal to

• A. 2
• B. 4
• C. $\frac{9}{2}$
• D. $\frac{11}{2}$

Answer 1

Answer: (B)

$\begin{vmatrix}0 & a-c & b-c \\ 0 & c-b & a-c \\ 1 & b & c\end{vmatrix}=0$
$(a-c)^2+(b-c)(b-a)=0 $
$a^2+c^2-a c+b^2-a b-b c=0 $
$(a-b)^2+(b-c)^2+(c-a)^2=0 $
$\therefore a=b=c \Rightarrow \text { triangle is equilateral. } $
$\therefore \sin ^2 A+\cos ^2 B +\tan ^2 C$
$\Rightarrow { }^{\circ}=4 $