Question

if A+b+c=90degrees then find tana tanb + tanb tanc + tanc tana

Answer 1

Answer:

Step-by-step explanation:

We have,

A+B+C = 90 (degree)

Now,

A+B = 90 - C ……(1)

Then,

tan(A+B) = tan(90 - C) (from (1))

or, tan(A+B) = tan(90 - C) ……(2)

As,

tan (x+y) = (tan x + tan y)/(1 - tan x tan y) ……(3)

and, tan (90 - x) = cot x ……(4)

Using (3) and (4) in (2), we get

(tan A + tan B)/(1 - tan A tan B) = cot C

or, (tan A + tan B)/(1 - tan A tan B) = 1/tan C

or, (tan C) (tan A + tan B) = (1 - tan A tan B)

or, (tan C tan A) + (tan B tan C) = (1 - tan A tan B)

or, tan A tan B + tan B tan C + tan C tan A = 1

Therefore, we have

tan A tan B + tan B tan C + tan C tan A = 1