(tan x- cot x)/( tan x + cot x)+ 2cos^2 x=1?

Answer 1

This is interesting....

Try this. First, change all tan(x) and cot(x) into sin(x)/cos(x) and cos(x)/sin(x) respectively. Then combine the fractions using common denominator which is cos(x)sin(x).

Eventually, the fraction will reduce to sin^2(x) -cos^2(x)

Plug this back into the original equation and you get:
sin^2(x) - cos^2(x) + 2cos^2(x) = 1
simply and you get:
sin^2(x) + cos^2(x) = 1
1 = 1

X can be any real number.....

Answer 2

(tan x- cot x)/( tan x + cot x)+ 2cos^2 x
= (sin x/cos x - cos x/sin x)/(sin x/cos x + cos x/sin x)+ 2cos^2 x
= ((sin^2 x - cos^2 x)/(sin x * cos x)) / ((sin^2 x + cos^2 x)/(sin x * cos x)) +2cos^2 x
= (sin^2 x - cos^2 x)/(sin^2 x + cos^2 x) + 2cos^2 x
= (sin^2 x - cos^2 x) / 1 + 2cos^2 x
= sin^2 x - cos^2 x + 2cos^2 x
= sin^2 x +cos^2 x
= 1