Question

tanπ/24.tan3π/24.tan5π/24.tan7π/24.tan9π/24.tan11π/24.ran16π/24=​

Answer 1

Answer:

Step-by-step explanation:

Tanπ/24.Tan3π/24.Tan5π/24.Tan7π/24.Tan9π/24.Tan11π/24.Tan16π/24

//We know that Tanθ = Cot(90 - θ) => Cot(π/2 - θ).

Tanπ/24 = Cot(π/2 - π/24) = Cot11π/24 ---  (1)

Tan3π/24 = Cot(π/2 - 3π/24) = Cot9π/24 --- (2)

Tan5π/24 = Cot(π/2 - 5π/24) = Cot7π/24 ---- (3)

Now substitute these values back in main equation.

=> Cot11π/24.Cot9π/24.Cot7π/24.Tan7π/24.Tan9π/24.Tan11π/24.Tan16π/24

//Re-arranging the above

=> (Cot11π/24*Tan11π/24) * (Cot9π/24* Tan9π/24) * (Cot7π/24.Tan7π/24) *  

     Tan16π/24

=> 1 * Tan16π/24   (∵ Tanθ * Cotθ = 1)

=> Tan2π/3

=> Tan120°

=> Tan(90°+30°)   (∵ Tan(90 + θ) = - Cotθ)

=> - Cot30°

=> - √3