Calculate the value of the expression 1/(tana+i)+1/(tana-i)+1/(cota+i)+1/(cota-i)

We'll add the first 2
fractions:

1/(tana+i)+1/(tana-i) = (tan a - i + tan a +
i)/(tana+i)*(tana-i)

We'll combine and eliminate like
terms:

1/(tana+i)+1/(tana-i) = 2*tan a/[(tan a)^2 -
i^2]

i^2 =
-1

1/(tana+i)+1/(tana-i) = 2*tan a/[(tan a)^2 + 1]
(1)

We'll add the next 2
fractions:

1/(cota+i)+1/(cota-i) = (cot a - i + cot a +
i)/(cota+i)*(cota-i)

We'll combine and eliminate like
terms:

1/(cota+i)+1/(cota-i) = 2*cot a/[(cot a)^2 + 1]
(2)

We'll add (1) + (2):

E =
2*tan a/[(tan a)^2 + 1] + 2*cot a/[(cot a)^2 + 1]

E =
[2*tan a/(tan a)^2 + 2*tan a + 2*cot a + 2*(tan a)^2/tan a]/[(tan a)^2 + 1]*[(cot a)^2 +
1]

E = (2/tan a + 2*tan a + 2/tan a + 2*tan a)/[(tan a)^2 +
1]*[(cot a)^2 + 1]

E = (4/tan a + 4*tan a)/[(tan a)^2 +
1]*[(cot a)^2 + 1]

E = [4 + 4*(tan a)^2]/tan a*[(tan a)^2 +
1]*[(cot a)^2 + 1]

E = 4*[1 + (tan a)^2]/tan a*[(tan a)^2 +
1]*[(cot a)^2 + 1]

E = 4/(cos a)^2/tan a*[1/(cos
a)^2]*[1/(sin a)^2]

E = 4*(sin a)^2/tan
a

E = 4*(sin a)^2*cos a/sin
a

E = 4*sin a*cos a

E =
2*2*sin a*cos a

We recognize the formula of sine of the
double angle:

E = 2*sin
(2a)

The requested value of the expression
is: E = 2*sin (2a).