UPSC CDS 2 2021 Math Paper

In a right angled triangle ABC, it is given that,
AB+BC=10(1+√3),∠B=90∘ and AC=20cmIn △ABC,
AB2+BC2=AC2
AB2+BC2=202
AB2+BC2=400 . . .(i)
Now, AB+BC=10(1+√3)
Squaring on both sides, we get
(AB+BC)2=[10(1+√3)]2
⇒AB2+BC2+2⋅AB⋅BC
=100(1+3+2√3)
⇒400+2AB⋅BC=100(4+2√3)
⇒2AB⋅BC=400+200√3−400
⇒AB⋅BC=100√3 . . .(ii)
Now, tan‌A+tan‌C
=‌

BC

AB

+‌

AB

BC

=‌

BC2+AB2

AB⋅BC

=‌

400

100√3

=‌

4

√3

Hence, tan‌A+tan‌C=‌

4

√3