UPSC CDS 2 2022 Math Paper

Pythagoras theorem: In a right-angled theorem,
Hypotenuse 2= Base 2+ Perpendicular ‌2
We have, AB=12‌cm,BC=9‌cm,AC=15‌cm 122+92=144+81=225=152
Hence, △ABC is a right-angled triangle in which ∠C=90∘
As we know, ∠A+∠B+∠C=180∘
⇒∠A+∠B=90∘‌‌(∵∠C=90∘)
⇒∠A=90∘−∠B
Now, the required value
sin2A+sin2B+sin2C‌
⇒sin2A+sin2(90∘−A)+sin290∘
⇒sin2A+cos2A+1‌[∵sin‌90∘=1]
⇒1+1=2‌[∵sin2θ+cos2θ=1]
∴sin2A+sin2B+sin2C=2