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Question
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Solution
Now `tan(theta + pi/3) = (tantheta + sqrt(3))/(1 - sqrt(3) tan theta)` and `tan(theta + (2pi)/3)`
= `(tantheta - sqrt(3))/(1 + sqrt(3) tan theta)`
So, `tan(theta + pi/3) + tan(theta + (2pi)/3)`
= `(tantheta + sqrt(3))/(1 - sqrt(3) tantheta) + (tan theta - sqrt(3))/(1 + sqrt(3) tantheta)`
= `((tan theta + sqrt(3))(1 + sqrt(3) tan theta) + (tan theta - sqrt(3))(1 - sqrt(3) tan theta))/(1 - 3tan^2theta)`
= `(tan theta + sqrt(3) + sqrt(3)tan^2theta + 3tantheta + tantheta - sqrt(3)tan^2theta - sqrt(3) + 3tantheta)/(1 - 3tan^2theta)`
= `(8tantheta)/(1 - 3tan^2theta)`
Given, `tantheta + tan(theta + pi/3) + tan(theta + (2pi)/3) = sqrt(3)`
⇒ `tan theta + (8tantheta)/(1 - 3tan^2theta) = sqrt(3)`
⇒ `(tantheta - 3tan^3theta + 8tantheta)/(1 - 3tan^2theta) = sqrt(3)`
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