UPSC CDS 2 2020 Math Paper

Consider statement 1 :
2cos2θ+cos‌θ−6=0
⇒2cos2θ+4‌cos‌θ−3‌cos‌θ−6=0
2‌cos‌θ(cos‌θ+4)−3(cos‌θ+4)=0
(cos‌θ+4)(2‌cos‌θ−3)=0
cos‌θ=−4 and cos‌θ=

3

2

We know that cos‌θ can never take value of -4 and

3

2

Hence, there is no value of θ for which the above equation holds.
Consider statement 2 :
A.M.≥G.M.

tan‌θ+cot‌θ

2

≥√tan‌θ×cot‌θ

tan‌θ+cot‌θ

2

≥√tan‌θ×

1

tan‌θ

tan‌θ+cot‌θ≥2
tan θ+cot‌θ is always greater than or equal to 2, where 0<θ<

π

2

.
Hence, only statement (2) is correct.