Question

if A+B=π/4,show that (cotA-1)(cotB-1)=2 and hence deduce that cot222.5°=√2+1.​

Answer 1

A+B=4π=4180=45°A+B=45°∴cot(A+B)=cot45°

∴cotBcotA−1(cotB+cotA)=1⇒cotB+cotA=cotBcotA=1⇒cotB+cotA−cotBcotA+1=0⇒cotBcotA−cotB−cotA−1=0⇒cotBcotA−cotB−cotA−1+2=0+2⇒cotBcotA−cotB−cotA+1=2⇒cotB(cotA−1)(cotA−1)=2⇒(cotA−1)(cotB−1)=2

Proved

Answer 2

Answer :-

A + B = π/4.

or, cot( A+B) = cot π/4.

or, ( cotA.cotB -1)/(cotB +cotA) = 1.

or, cotA.cotB -1 = cotB + cotA.

or, cotA.cotB -cotB-cotA = 1.

or, cotB.(cotA - 1) - (cotA-1) = 1+ 1.

or, (cotA -1).(cotB -1) = 2. Proved.