Question

The value of `tan  π/5 + 2 tan  (2π)/5 + 4 cot  (4π)/5` is ______.

Options

• `cot  π/5`

• `cot  (2π)/5`

• `cot  (4π)/5`

• `cot  (3π)/5`

Answer 1

The value of `tan  π/5 + 2 tan  (2π)/5 + 4 cot  (4π)/5` is `underlinebb(cot  π/5)`.

Explanation:

`tan  π/5 + 2 tan  (2π)/5 + 4 cot  (4π)/5`

= `(sin  π/5)/(cos  π/5) + 2 (sin  (2π)/5)/(cos  (2π)/5) + 4 (cos  (4π)/5)/(sin  (4π)/5)`

= `(sin  π/5)/(cos  π/5) + 2(sin  (2π)/5)/(cos  (2π)/5) + (4(cos^2  (2π)/5 - sin^2  (2π)/5))/(2 sin  (2π)/5 cos  (2π)/5)`

= `(sin  π/5)/(cos  π/5) + 2(sin  (2π)/5)/(cos  (2π)/5) + (2(cos^2  (2π)/5 - sin^2  (2π)/5))/(sin  (2π)/5 cos  (2π)/5)`

= `(sin  π/5)/(cos  π/5) + 2[(sin^2  (2π)/5 + cos^2  (2π)/5 - sin^2  (2π)/5)/(cos  (2π)/5 sin  (2π)/5)]`

= `(sin  π/5)/(cos  π/5) + (2 cos  (2π)/5)/(sin  (2π)/5)`

= `(sin  π/5)/(cos  π/5) + (2(cos^2  π/5 - sin^2  π/5))/(2sin  π/5 cos  π/5)`

= `(sin^2  π/5 + cos^2  π/5 - sin^2  π/5)/(sin  π/5 cos  π/5)`

= `(cos^2  π/5)/(sin  π/5 cos  π/5)`

= `cot  π/5`