UPSC NDA Math Model Paper 9

Since, A, B and C are the angles of a triangle, we have A + B + C = π.
Now, cos‌2 A + cos‌2 C = sin‌2 B
⇒ cos‌2 A + (cos‌2 C - sin‌2 B) = 0
⇒ cos‌2 A + cos (C + B) cos (C - B) = 0
⇒ cos‌2 A + cos (π - A) cos (C - B) = 0
⇒ cos‌2 A - cos A cos (C - B) = 0
⇒ cos A [cos A - cos (C - B)] = 0
⇒ cos A [cos (π - (B + C)) - cos (C - B)] = 0
⇒ -cos A [cos (B + C) + cos (C - B)] = 0
⇒ -2 cos A cos B cos C = 0
⇒ cos A = 0 OR cos B = 0 OR cos C = 0
⇒A=

π

2

‌OR‌B=

π

2

‌OR‌C=

π

2

∴ The triangle is right-angled.