Trigonometry

Question

Given that cotθ=k and that θ is acute, express tan3θ in terms of k.

 

Ans: (3k²-1)/(k)(k²-3)

 

 

 

 

Thank you :)

Answer 1

cotθ=k

 

tanθ=1/k

 

tan3θ=tan(θ+2θ)

        =tanθ+tan2θ

          1-tanθtan2θ

 

       =tanθ+2tanθ/1-tan²θ

          1-tanθ.2tanθ/1-tan²θ

 

        =tanθ(1-tan²θ)+2tanθ

         1-tan²θ-2tan²θ

 

        =tanθ-tan³θ+2tanθ

           1-3tan²θ

 

        =3tanθ-tan³θ

           1-3tan²θ

 

       =  3(1/k)-(1/k)³

            1-3(1/k)²

 

       =(3k²-1)/k³

         ( k²-3)/k²

 

       =(3k^²-1)/(k)(k²-3)