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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find the value of 1/(tan a + i) +1/(tan a - i)+1/(cot a + i)  + 1/(cot a - i)

1/(tan a + i) +1/(tan a - i)+1/(cot a + i)  + 1/(cot a - i)

=> 1/(tan a + i) +1/(tan a - i)+1/((1/ tan a) + i)  + 1/(1/ tan a) - i)

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

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

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

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

=> [(1 + i*tan a)(tan a - i) + (1 - i*tan a)(tan a + i)]/(tan a - i)(tan a + i)

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

=> [tan a + tan a + tan a + tan a]/((tan a)^2 + 1)

=> (4*tan a)/(1 + (sin a)^2/(cos a)^2)

=> 4*(sin a/cos a)*(cos a)^2/((cos a)^2 + (sin a)^2)

=> 4*sin a* cos a

=> 2* sin 2a

The value of the expression is 2* sin 2a

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

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