Question

If a+b= 225 prove that (1+cota)(1+cotb) = 2cotacotb, please answer the question with steps.

Answer 1

Answer:

(1+cota)(1+cotb) = 2cotacotb

cotA/(1+cotA) *cotB/(1+cotB) =1/2

_____________

A + B = 225 = B = 225 - A

= B = 180 + 45 - A = B = 180 + (45 - A) (As 225 = 180 + 45)

Putting value of B in question we get

cot A/(1 + cot A) * cot {180 + ( 45 - A)} / [ 1 + cot {180 + (45 - A)}

= cot A/(1 + cot A) * cot (45 - A) / [1 + cot (45 - A)] [As cot (180 + A) = cot A]

= cot A/(1 + cot A) * [ {cot 45cot A + 1 / cot A - cot 45} / {1 + (cot 45 cot A + 1) / (cot A - cot 45)}]

[As cot (A - B) = (cot B cot A + 1) / (cot B - cot A) ]

= cot A/(1 + cot A) * [ { cot A + 1 / cot A - 1} / {1 + ( cot A + 1) / (cot A - 1)}] (As cot 45 = 1)

= cot A/(1 + cot A) * [ { cot A + 1 / cot A - 1} / { (cot A - 1+ cot A + 1) / (cot A - 1)}] [ Taking L.C.M.]

= cot A/(1 + cot A) * [{cot A + 1 / cot A - 1} / { (2 cot A) / (cotA - 1)}]

= cot A/(1 + cot A) * (cot A + 1) / (cot A - 1) * (cot A - 1) / (2 cot A)

=1/2