Question

If A+B=45 then prove that (cotA-1)(cotB-1)=2
and find the value of COT22*1/2°

Answer 1

A+B=45°
∴, cot(A+B)=cot45°
or, (cotB+cotA)/(cotBcotA-1)=1
or, cotB+cotA=cotBcotA-1
or, cotB+cotA-cotBcotA+1=0
or, cotBcotA-cotB-cotA-1=0
or, cotBcotA-cotB-cotA-1+2=0+2 (Adding 2 both sides)
or, cotBcotA-cotB-cotA+1=2
or, cotB(cotA-1)-1(cotA-1)=2
or, (cotA-1)(cotB-1)=2 (Proved) ------------------(1)
22 1/2°+22 1/2°=45°
∴, cot(22 1/2°+22 1/2°)=cot45°
∴, (cot22 1/2°-1)(cot22 1/2°-1)=2 [Using (1)]
or, (cot22 1/2°-1)²=2
or, cot22 1/2°-1=√2
[∵, 22 1/2° is in the first quadrant we can neglect the negative sign]
or, cot22 1/2°=√2+1