# For all possible values of variable x, what is the result of the sum? (1+sin^2x)/(2+cot^2x)+(1+cos^2x)/(2+tan^2x)=

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We know, from Pythagorean identity, that:

1 + (tan x)^2 = 1/(cos x)^2

1 + (cot x)^2 = 1/(sin x)^2

We'll re-write the denominators:

1 + 1 + (cot x)^2 = 1 + 1/(sin x)^2

2 + (tan x)^2 = 1 + 1 + (tan x)^2 = 1 + 1/(cos x)^2

We'll re-write the expression:

[1+(sinx)^2]/(2+(cot x)^2)+[1+(cos x)^2]/(2+(tan x)^2)=

[1+(sinx)^2]/(1 + 1/(sin x)^2) + [1+(cos x)^2]/(1 + 1/(cos x)^2) =

[1+(sinx)^2]/[((sinx)^2 + 1)/(sin x)^2] + [1+(cos x)^2]/[((cos x)^2 + 1)/(cos x)^2] =

(sin x)^2 + (cos x)^2 = 1

**The required value of the given sum is S = 1.**