UPSC NDA I 2025 Mathematics Paper

Letp =sin‌35∘,q=sin‌25∘ and r=sin‌(−95∘)
We need to find the value of p+q+r.
First, we know that:
sin‌(−95∘)=−sin‌(95∘).
Thus, the expression p+q+r becomes:
p+q+r=sin‌35∘+sin‌25∘−sin‌95∘.
Using the identity for the sum of two sine functions:
sin‌A+sin‌B=2sin‌(‌

A+B

2

)‌cos(‌

A−B

2

).
Substitute A=35∘ and B=25∘
sin‌35∘+sin‌25∘=2sin‌30∘‌cos‌5∘.
Since sin‌30∘=‌

1

2

, we get:
sin‌35∘+sin‌25∘=cos‌5∘.
Now, we know that:
sin‌95∘=cos‌5∘.
Therefore:
p+q+r=cos‌5∘−cos‌5∘=0
∴ The value of p+q+r is 0.