UPSC CDS I 2025 Mathematics Paper

For Statement I:
(cosec‌α− secα)=1∕sin‌α−1∕cos‌α
⇒ Taking LCM:
(cosec‌α− secα)=(cos‌α−sin‌α)∕(sin‌α×cos‌α)
Since cos‌α and sin‌α are both positive in the first quadrant, the numerator ( cos‌α−sin‌α ) can be positive, zero, or negative depending on the specific value of α. Thus, Statement I is not always true.
For Statement II:
(tan‌α−cot‌α)=(sin‌α∕cos‌α)−(cos‌α∕sin‌α)
⇒ Taking LCM:
(tan‌α−cot‌α)=(sin‌2α−cos2α)∕(sin‌α×cos‌α)
In the first quadrant, sin‌α and cos‌α are positive. However, sin‌2α−cos2α is negative because cos‌α>sin‌α for α in the first quadrant. Hence, (tan α−cot‌α ) is always negative in the first quadrant.