if a b c 0 , prove that cot-1((ab+1)/(a-b)) + cot-1((bc+1)/(b-c)) + cot-1((ca+1)/(c-a))=pi - Maths - Inverse Trigonometric Functions

Question

if a>b>c>0 , prove that
cot-1((ab+1)/(a-b)) + cot-1((bc+1)/(b-c)) +
cot-1((ca+1)/(c-a))=pi

Tanveer Sofi · Accepted Answer

We have to provecot-1ab+1a-b+cot-1bc+1b-c+cot-1ca+1c-a=πWe know thattan-11x=cot-1x,if x>0-π+cot-1x,if x<0⇒cot-1x=tan-11x,if x>0π+tan-11x, if x<0Therefore we can writecot-1ab+1a-b=tan-1a-b1+ab As, a>b>0⇒a-b>0 and 1+ab>1⇒a-b1+ab>0cot-1bc+1b-c=tan-1b-c1+bc As, b>c>0⇒b-c>0 and 1+bc>1⇒b-c1+bc>0andcot-1ca+1c-a=π+tan-1c-a1+ca As, c0⇒c-a<0 and 1+ca>1⇒c-a1+ca<0Hence, we can writecot-1ab+1a-b+cot-1bc+1b-c+cot-1ca+1c-a=tan-1a-b1+ab+tan-1b-c1+bc+π+tan-1c-a1+ca=tan-1a-tan-1b+tan-1b-tan-1c+π+tan-1c+tan-1a As, tan-1x-y1+xy=tan-1x-tan-1y=πHence proved

if a>b>c>0 , prove that cot-1((ab+1)/(a-b)) + cot-1((bc+1)/(b-c)) + cot-1((ca+1)/(c-a))=pi

if a>b>c>0 , prove that
cot^-1((ab+1)/(a-b)) + cot^-1((bc+1)/(b-c)) + cot^-1((ca+1)/(c-a))=pi