If-tan-3A-tan-A-k-then-sin-3A-sin-A-is-equal-to-

Question Number 73706 by Faradtimmy last updated on 15/Nov/19

If \frac{\tan 3 A}{\tan A} = k, then \frac{\sin 3 A}{\sin A} is equal to

Answered by Tanmay chaudhury last updated on 15/Nov/19

t a n A = t

\frac{3 t - t^{3}}{t (1 - 3 t^{2})} = k \to k - 3 {k t}^{2} = 3 - t^{2}

t^{2} (1 - 3 k) = 3 - k {t a n}^{2} A = \frac{3 - k}{1 - 3 k}

{c o t}^{2} A = \frac{1 - 3 k}{3 - k} \to {c o s e c}^{2} A = \frac{1 - 3 k + 3 - k}{3 - k} = \frac{4 - 4 k}{3 - k}

{s i n}^{2} A = \frac{3 - k}{4 - 4 k}

\frac{s i n 3 A}{s i n A}

= 3 - 4 {s i n}^{2} A

= 3 - 4 (\frac{3 - k}{4 - 4 k}) = \frac{12 - 12 k - 12 + 4 k}{4 - 4 k} = \frac{- 8 k}{4 - 4 k} = \frac{2 k}{k - 1}