Question

In triangle $ABC$ if ${\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C=1$,then identify the type of triangle

Answer 1

Given, ${\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C=1$

rearranging

$\Rightarrow {\cos }^{2}B-{\cos }^{2}C=1-{\cos }^{2}A={\sin }^{2}A$

using trigonometric identities
$\Rightarrow \sin (B+C).\sin (C-B)={\sin }^{2}A$

sum of angles of triangle is $\pi$
$\Rightarrow \sin (\pi -A).\sin (C-B)={\sin }^{2}A$
$\Rightarrow \sin A.\sin (C-B)={\sin }^{2}A\Rightarrow \sin (C-B)=\sin A$
$\Rightarrow C-B=A$

$\Rightarrow A+B=C$

$\Rightarrow C=\frac{\pi }{2}$
Hence the triangle will be right angled.