If ( tan 3 A/ tan A)=a, then ( sin 3 A/ sin A) is equal to

Question

If ( tan 3 A/ tan A)=a, then ( sin 3 A/ sin A) is equal to (A) (2 a/a+1) (B) (2 a/a-1) (C) (a/a+1) (D) (a/a-1)

Tardigrade Admin · Accepted Answer

Given that ( tan 3 A/ tan A)=a ⇒ (3 tan A- tan 3 A/ tan A(1-3 tan 2 A))=a ⇒ 3- tan 2 A=a-3 a tan 2 A ⇒ tan 2 A(3 a-1)=a-3 ⇒ tan A=± √(a-3/3 a-1) Now, <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/3ef7d17bcab2bf30b57aece21f250718-.png" /> ( sin 3 A/ sin A)=(3 sin A-4 sin 3 A/ sin A) =3-4 sin 2 A=3-4((a-3/4(a-1))) =(3 a-3-a+3/(a-1))=(2 a/(a-1))

Q. If $\frac{t a n 3 A}{t a n A} = a$ , then $\frac{s i n 3 A}{s i n A}$ is equal to

$\frac{2 a}{a + 1}$

$\frac{2 a}{a - 1}$

$\frac{a}{a + 1}$

$\frac{a}{a - 1}$

Q. If tanAtan3A​=a, then sinAsin3A​ is equal to

a+12a​

a−12a​

a+1a​

a−1a​

Given that tanAtan3A​=a ⇒tanA(1−3tan2A)3tanA−tan3A​=a ⇒3−tan2A=a−3atan2A ⇒tan2A(3a−1)=a−3 ⇒tanA=±3a−1a−3​​ Now, sinAsin3A​=sinA3sinA−4sin3A​ =3−4sin2A=3−4(4(a−1)a−3​) =(a−1)3a−3−a+3​=(a−1)2a​

Q. If $\frac{t a n 3 A}{t a n A} = a$ , then $\frac{s i n 3 A}{s i n A}$ is equal to

$\frac{2 a}{a + 1}$

$\frac{2 a}{a - 1}$

$\frac{a}{a + 1}$

$\frac{a}{a - 1}$