UPSC CDS 1 2022 Math Paper

Given:
cos(x+y)=0
sin(x−y)=‌

1

2

Concept Used:
sin‌A=sin‌B then A=B
cos‌A=cos‌B then A=B
Where 0≤A,B≤90∘
Formula Used:

cos‌90∘=0,sin‌30∘=1∕2,cot‌90∘=0
Calculation:
We have
cos(x+y)=0
⇒cos(x+y)=cos‌90∘
⇒cos(x+y)=cos‌90∘
⇒x+y=90∘
And sin(x−y)=‌

1

2

⇒sin(x−y)=sin‌30∘
⇒x−y=30∘
On adding equation (i) and (ii), we get
2x=120∘
⇒x=60∘
Now, From (i)
60∘+y=90∘
⇒y=30∘
Now, We have to find cot(2x−y)
So, cot(2×60∘−30∘)=cot‌90∘=0
∴ The value of cot(2x−y) is 0 .